There wasn't any Austro-Hungarian Empire. After 1867 there was the dual realm of Austria-Hungary, a sort of confederation of only two parts. It consisted of Austria, more or less, and of Hungary, more or less, and the two parts were independent in most respects but had a few common institutions. Thus there were three governments, the Austrian government, the Hungarian government, and the common or joint government.

The Austrian government more or less ruled the Austrian part, the Hungarian government more or less ruled the Hungarian part, and there was a common government with input from both parts. And probably various power struggles between the two parts to control the common government.

The Austrian part had been an empire since 1804 and sometimes included Hungary (more often in the Austrian opinion than in the Hungarian opinion), while the Hungarian part had been a kingdom since about 1000 AD.

So it is proper to call the combined realm Austria-Hungary, which was its short official name. Or you could call it the Empire of Austria and Kingdom of Hungary.

But calling it the Austro-Hungarian Empire is just plain wrong. An Austro-Hungarian Empire was never officially decreed to exist. Nobody ever used the title of Austro-Hungarian Emperor or Emperor of Austro-Hungary.

The Habsburgs had developed - much like other European powers at the time - a quite centralised rule in the 18th century, under monarchs like Maria Theresia and her son Joseph. Vienna was the place-to-be and the other parts of the realm who had had individual rules and regulations in place since centuries saw those abolished and replaced with standardized ones.

Especially lands like Hungary, Bohemia and Tyrol with strong self-representation traditions had a problem with that and Hungary declared itself independent in the revolution year of 1848. Since Hungary was such a big problem for the Habsburgs, they allowed the formal independence from Austria with only the monarch being identical and a few shared government ministries (Foreign Affairs, Military and Trade). So from 1866 onwards you had a strong Hungarian influence over all "Hungarian" lands (incl. Slovakia, Croatia and Transsylvania). There never was Hungarian influence on "German-Austrian" lands though.