Have you heard of this little country called Finland?
A good discussion could be had over to what extent Finland can be seen as a successor state of Tsarist/non-Bolshevik Russia. There are certain elements that point to this direction - for example, of all the states that were formed in the areas of the former Russian Empire, Finland had the strongest legal and organizational continuity from the Tsarist days. Arguably we still have laws in place that were originally decreed by the Tsar of Russia. And of course many of the early leaders of independent Finland and its armed forces had been in leading positions in the Finnish Grand Duchy as politicians or bureaucrats under the Tsarist system, or military officers in the Tsarist military like Mannerheim, Löfström, Nenonen, von Schoultz and a number of others.
In 1918, after the Finnish Reds (who had strong Russian revolutionary support) had been beaten in the civil war, Finland was arguably "White" and anti-Bolshevik. But then this independent Finland most emphatically was not a "White Russian" state as such, but could be best described as a Finnish bourgeois state. Actual Russian influence on the Finnish White side was negligible, the number of ethnically and culturally Russian White emigrants in Finland was small and they did not have any real power. Even the Finnish-born officers who had a background in the Tsarist military were mostly sidelined from leading positions in the 20s and 30s. And then of course in independent Finland also the left was soon reintegrated into the political system after the civil war, in the 20s and 30s making the state less "White" than it still looked like after the brief royalist side track in 1919.
The various continuities from the Tsarist days were mostly circumstantial and essentially based on historical inertia - in terms of the development of laws, the political system and governance, the Republic of Finland was not formed via a revolution, but through an evolutionary process beginning with the Finnish Grand Duchy (which in turn had its roots in the Swedish system pre-1809).