1. The Space Shuttle flew a large number of lengthy, classified missions.
2. The Space Shuttle suffered from many costly design compromises due to its excessively large hangar bay.
Given #1 and #2, there are four likely kinds of classified missions that the Space Shuttle undertook.
1. Stealing Russian military satellites.
2. Returning American military satellites to Earth.
3. Launching American military satellites into unpredictable orbits.
4. Refueling American military satellites, which allowed them to change to a new, unpredictable orbit.
It's entirely possible that the Space Shuttle only engaged in some of these tasks... But it's difficult to conclude that given the very odd design of the space shuttle, and the very lengthy missions that it undertook, that no satellite servicing/orbital shenanigans took place.
Yes, satellites can change orbits using on-board fuel reserves, but due to the tyranny of the rocket equation, those reserves are limited - and are unlikely to allow a very large inclination change (Inclination changes are incredibly fuel-expensive.)