Because I didn't thoroughly respond to the OP, I'll do so now.
I decided to look up corresponding articles on the Wiki in order to see whether more empires expanded to the east or west, tallied them up, and found out that the numbers were roughly equal. Although the list below is not comprehensive, I tried to find the ones that substantially expanded east or west relative to the other directions, and only choose the ones that were at the least very influential regional powers. I also combined the ones that were substantially similar in terms of culture and area covered. On the other hand, I omitted the maritime colonial powers (Portugal, Spain, Netherlands, France, and Britain), because unlike most of the others, in terms of east/west, they ended up spanning most of the globe. Here's what I ended up with, in rough chronological order:
West to East:
Macedonian: The earliest major entity to do so, and stated in the OP. It originated in Macedonia, north of the Greek city-states, taking them over, then expanded into Persia.
Carthaginian (Both): Although its founders originated from Phoenicia, the state itself originated in what is now Tunisia, and expanded into Libya and Sicily.
Qin (West)/Han/Tang (Both): All three originated from Shaanxi, and managed to take over the rest of China Proper.
Roman/Byzantine (Both): Rome expanded from Italy into the Balkans, Greece, Libya, Egypt, and Anatolia. It also briefly expanded into Mesopotamia under Trajan, its rule was short-lived. On the other hand, Byzantium retained the eastern parts of the preceding state, but it was often forced to reconquer some of its lost territories in Anatolia.
Goguryeo/Balhae (Both): While the former originated from Jolbon, in southwest Manchuria, and its successor was founded on Dongmo Mountain, in southeast Manchuria, both managed to expand westward into the Sea of Japan, and culturally influence the Mohe tribes living in the region.
Frankish/Carolingian: Both originated around Western France and expanded their dominions into what is now Germany and Eastern Europe.
Islamic Caliphate (Both): The state originated in Arabia, and spread rapidly into Mesopotamia and Persia.
HRE/Austria: The former was mostly composed of the eastern regions of the former Carolingian state, and its successor expanded into Hungary and the Balkans.
Majapahit (Both): Originally established in Java and projected its power into Sulawesi, Timur, Papua, and other islands to the east.
Burma: Expanded into what is now Thailand.
Polish-Lithuanian Commonwealth: Both expanded east before they were joined in a union.
Swedish: Took over Finland and the Baltics.
Russian: Initially explored, then eventually conquered Siberia.
Siam (Both): Controlled what is now Laos, Cambodia, and parts of Vietnam.
First French: Although technically not relevant to this thread, it managed to take over the Low Countries and Italy, while pressuring Prussia and Austria into allying with it.
Total: 15
East to West:
Assyria: The first significant state to do so. Originated in central Mesopotamia, and expanded into northern Mesopotamia, Anatolia, the Levant, and Egypt.
Median/Achaemenid: The latter is also known as Persia. Both originated from what is now Iran, and eventually took over Assyria's former possessions.
Carthaginian (Both): Conquered western North Africa, Sardinia, Corsica, and southern Spain.
Parthian/Sassanid: With the exception of Egypt, both expanded in a similar manner as their predecessor, Persia, and clashed with Rome over Anatolia and Mesopotamia.
Han/Tang (Both): Both set up commanderies in what is now Xinjiang.
Xiongnu/Hunnic/Rouran/Turkish/Uyghur: Although these states maintained distinct identities from one another, with the exception of the Huns, they all expanded westward from their original bases near what is now Mongolia.
Roman/Byzantine (Both): The former controlled what is now France and England, while the latter briefly retook western North Africa, Italy, and southern Spain.
Goguryeo/Balhae (Both): Both expanded into the Liaodong Peninsula, while the former briefly occupied what is now Beijing in the early 5th century.
Islamic Caliphate (Both): Took over North Africa and Spain.
Pala/Eastern Ganga: Both were founded in Bengal, and conquered most of Northern India and parts of the south as well.
Khmer: Established in modern day Cambodia, and controlled what is now Laos and Thailand.
Srivijaya (West)/Majapahit (Both): Both occupied Sumatra and Malaysia.
Mongol: After disparate tribes were unified in what is now Mongolia, it eventually managed to span most of Eurasia, with the exceptions of Siberia, India, Southeast Asia, Arabia, and Western Europe, although it was fragmented among Genghis Khan's sons soon after his death.
Ottoman: From its base in Anatolia, it managed to expand into Greece, the Balkans, the Levant, and North Africa.
Timurid: Originated from what is now Uzbekistan, and spread into Mesopotamia and the Black Sea.
Qing: Although it reunified China Proper, it initially established its base in southern Manchuria, and conquered Xinjiang and Tibet.
Siam (Both): Controlled Burma.
Total: 17
