Are we talking about the true Habsburg lineage that ended in 1780 with the death of the Empress Maria Theresia?

When the ambitious Rudolf IV (of Privilegium Maius fame) died in 1365 he left his two younger brothers in joint rulership of his indivisible lands (Rudolfinische Hausordnung)... they did not get along though and in 1379, without outside pressure and against the explicit decree, they divided the family lands between them in the Treaty of Neuberg.
Albert was to keep the ancestral lands of Austria proper (Österreich ob und nid der Enns, now more or less the provinces of Upper and Lower Austria) while Leopold should inherit Styria, Carinthia and Carniola, Tyrol, the patchwork of holdings in Further Austria (now Switzerland mostly) including the ancient family seat and sme coastal area in Friuli.

This effectively divided the House into two sub-branches, the Albertinian and the Leopoldinian line. And actually even further when the Leopoldinian holdings were further split between two of the original Leopold's sons. These splintered powerbases of the Habsburg lines would remain until in 1490 Maximilian I. (of the HRE) eventually could unite all family possessions.

While relations were sometimes strained between those two branches there was never any real animosity as far as I could recall. Rather they were united against the constant Ottoman danger close to their borders. And in fact, when Albrecht IV. the patient (from the Albertinian side) died and left the lands to his underaged son the Leopoldinians provided themselves as regents until he was of age.

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Then in 1521/22 the Habsburgs part again with the Treaties of Worms and Brussels into the Austrian and the Spanish line. Though here relations are pretty good all the way through to the extinction of the Spanish Habsburgs in 1700 since both parties are willing and there is very little conflict of interest between the two.

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And once again with the parting. When Ferdinand I. died in 1564 he had his dominion split between his three sons with the Ferdinandeische Hausordnung because he distrusted his firstborn, Maximilian, and was afraid he would convert to Protestantism and could otherwise force it upon all his lands. This despite the fact that one of the motivations for the original forging of the Privilegium Maius (in the mean time having been confirmed by a Habsburg Emperor), which made Austria an Archduchy with special status and privileges, was to make the family lands indivisible and thus the power base stronger.

Maximilian, who was still his heir both personal and as Holy Roman Emperor, would inherit the core lands of Austria proper as well as the crowns of Hungary and Bohemia. The second son and his personal favourite, Ferdinand, got the hardcore Catholic Duchy of Tyrol (Austria Superior) and the youngest, Charles, the Duchies of Styria and Carinthia-Carniola as well as the coast (together Inner Austria).

However, all three could and did use the same full family title and coat of arms and since both the first and second line failed to produce an heir all parts fell back to Charles and his descendants in the 1620. Once again I can't think of any conflict worth mentioning.

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There's probably much more to say here, especially if you want to include the more well-known successor House of Habsburg-Lorraine but I am really quite tired now so I'll have to come back to you on that. Also, I'm on vacation for the remaining part of the week and thus away from most of my library. So while you are very welcome to ask follow-up questions or anything else just a heads up to expect some delays, especially when it comes to the Medieval part.