Challenge: Holy Roman Empire actually recognized as the Roman Empire

This is easier/likelier if the HRE comes to dominate Greater Italy instead of Greater Germany.

At the latest that could happen under the Sfaufer (Hohenstaufen) Henry VI and Frederick, a perfect early candidates would be Otto II and Otto III, however there are many periods in between, which aren't suited for this too. Moreover they could only dominate one half with the added resources of the other half, neither the Ottonians or the Staufer were able to solely relay on their position in Italy for this.
 
At the latest that could happen under the Sfaufer (Hohenstaufen) Henry VI and Frederick, a perfect early candidates would be Otto II and Otto III, however there are many periods in between, which aren't suited for this too. Moreover they could only dominate one half with the added resources of the other half, neither the Ottonians or the Staufer were able to solely relay on their position in Italy for this.
Perhaps not, but a Byzantine marriage might help.
 
Perhaps not, but a Byzantine marriage might help.
Not especially; Romanness isn't inherited that way, and Byzantine succession wasn't strictly hereditary anyway. The Russians claimed to be the "Third Rome" on the basis of their Byzantine marriage, but no one other than the Russians took that any more seriously than they took the HRE. Which gets us back to the question of "who is doing the recognizing."
 
5. The Byzantines did call themselves 'Romanoi", and the first Turkish successor state parked on what had been Byzantine territory was called "Rum", but then what was this mixture of Greeks, Armenians, Slaves, and what-not going to call themselves? There was no reason for the people living in the Kingdoms of Germany and Italy not to continue to call themselves "Germans" and "Italians".

I'm going to push back here a little bit. The Byzantines certainly saw themselves as Roman, and their histories are replete with analogies to previous Roman Emperors. So it wasn't that they lacked an alternative title.
 
Not especially; Romanness isn't inherited that way, and Byzantine succession wasn't strictly hereditary anyway. The Russians claimed to be the "Third Rome" on the basis of their Byzantine marriage, but no one other than the Russians took that any more seriously than they took the HRE. Which gets us back to the question of "who is doing the recognizing."
No, the Russians don't count, but someone holding Italy who is well connected to Constantinople is probably not a Russian.
 
It would be a lot easier for nobody to dispute if you just nipped the Roman Empire in the bud. Have it overrun by Avars, Persians, Arabs, etc. early on. Nobody will nitpick if there's no state with any real continuity left as Roman Empire.
 
Perhaps not, but a Byzantine marriage might help.

Otto II was married with Theophanu a niece of the Byzantine Emperor John I. King of the Romans Philip of Swabia, uncle of Frederick II and thus a brother of Henry VI, he was married to Irene Angelina, but was assassinated before he could be crowned Emperor by the Pope. A frankly after the Byzantine Empire became the de facto Constantinople city state, the value of a Byzantine marriage had decreased a lot. Sure this leaves centuries to increase that number, but after a certain point strategic marriages with France or England could be more advantageous than an Byzantine marriage.
 
Otto II was married with Theophanu a niece of the Byzantine Emperor John I. King of the Romans Philip of Swabia, uncle of Frederick II and thus a brother of Henry VI, he was married to Irene Angelina, but was assassinated before he could be crowned Emperor by the Pope. A frankly after the Byzantine Empire became the de facto Constantinople city state, the value of a Byzantine marriage had decreased a lot. Sure this leaves centuries to increase that number, but after a certain point strategic marriages with France or England could be more advantageous than an Byzantine marriage.
True, but there would be symbolic value in ruling Italy and capturing the Golden Horn.
 
True, but there would be symbolic value in ruling Italy and capturing the Golden Horn.

Symbolism in general is trumped by the real pressing political matters. For instance even in the day of the Valois dukes of Burgundy, the idea of a crusade was entertained by their court, but of the fine proud and pious words aside it never happened. So how the nice ruling the Golden Horn might be, that might stay a dream, ruling over Italy is a lot more achievable though.
 
Symbolism in general is trumped by the real pressing political matters. For instance even in the day of the Valois dukes of Burgundy, the idea of a crusade was entertained by their court, but of the fine proud and pious words aside it never happened. So how the nice ruling the Golden Horn might be, that might stay a dream, ruling over Italy is a lot more achievable though.
True. But does one preclude the other?
 
What if the HRE was a "Holy Roman Empire of the Italian Nation" as opposed to a "Holy Roman Empire of the German Nation"?
 
These points will be fairly random:

1. It was recognized as the Roman Empire. The Holy Roman Emperor was crowned as such as the Pope, recognized by the Byzantine/ Eastern Roman Emperor as his colleague, etc. By the way, official government documents were printed in Latin down to 1806. I don't get what the "what if" means.

2. The term "Holy Roman Empire" was an invention of Barbarossa for propaganda purposes that for some reason stuck.

3. The term "Roman" really applied to the rulers, not the population. This is confusing. Charlemagne made a bid to be recognized as Byzantine emperor on the grounds that the position was at the time occupied by a woman, in other words vacant. The Pope crowned him (no actual Roman Emperor, East or West, was ever crowned by the Pope). Once the Byzantines replaced a woman with a man, they struck a deal that Charlemagne would be recognized as the colleague of the Roman Emperor in Constantinople, in exchange for evacuating some cities he had occupied that were aligned with them at the time. As Bury pointed out, this wasn't as big a deal for them as it seems to us. After 285, the Roman Empire usually had two Emperors, one in the East and one in the West, and the Byzantines themselves often had more than one person legally holding the title. After Charlemagne died, the title passed to his son Louis and grandson Lothair, and then to whoever had the best claim to be King of (usually anarchic) Italy.

4. After the Treaty of Verdun, Charlemagne's empire was partitioned, and as stated above whoever had the best title to the Italian spin-off was generally recognized as Roman Emperor. After becoming King of Germany, the Saxons conquered Italy and became King of Italy as well. That got them the title. Some, not all, of them chose to make a big deal of it. The Kingdoms of Italy and Germany continued as entities and the Holy Roman Emperor was usually crowned as such, often before he officially assumed the grander title. Other than the Emperor himself, there were no common institutions between the two kingdoms.

5. The Byzantines did call themselves 'Romanoi", and the first Turkish successor state parked on what had been Byzantine territory was called "Rum", but then what was this mixture of Greeks, Armenians, Slaves, and what-not going to call themselves? There was no reason for the people living in the Kingdoms of Germany and Italy not to continue to call themselves "Germans" and "Italians".

If you want people in Western Europe, other than inhabitants of Rome itself, to continue to call themselves "Romans", you need a completely different set-up than the historical Holy Roman Empire. You also need a POD before 1000.

There are three ways to do this, and they are all unlikely. The first is that is that the East Roman Emperor himself sets up a Western colleague and essentially revives the Western Roman Empire, and it somehow becomes more successful than IOTL and expands. The time to do thiw was in the second half of the fifth century, when the idea was considered, but maybe it could be done as late as the reign of Constans Pogonatos. No Arab conquests would help enormously.. The second is that Charlemagne and Louis the Pious both actually want to and are able to keep the empire united, maybe in Constantinople they make Charlemagne the Emperor when Irene is removed, and they import alot of people who can read and write from Constantinople who can staff a bureaucracy. The third is that Otto III, who among Holy Roman Emperors took the title the most seriously, both lives longer and succeeds in reviving the corpse.

The title may have passed down through Charlemagne's descendants, but the recognition by the Roman Emperors died with Charlemagne as that is with whom the Pax Nikophorii was made. Your point about there being a dyarchy is correct, however, ever since the Romans moved their capital to Nova Roma (Konstantinoupolis) there has been a senior and junior emperor. The empire focused east as that was the wealthiest, most cultured (in their eyes) area of the Politeia. Old Rome was a quaint reminder of old times and a decreasing population. The Senior emperor always resided in Constantine's City, while the junior emperor went nilly willy in the Occident. The Junior emperor was approved by the senior emperor, G-d, the army, and the senatus popolusque romana. They had real power, but little wealth and resources due to the mercenary makeup of the army and the later movement of the Germans. The HRE is not Roman in that it is not of the Roman Empire. The Emperors in Nova Roma did not create Charlemagne as a junior emperor (and he certainly did not want to be a junior to anyone). The HRE was not a res publica and was treated as the personal domain of the Carolingians and their vassals (it is why the Romans called them and the westerners barbarians).
 
Top