Depends on the deck, but generally speaking, the capstan motor never changes direction. On cassette mechanisms without auto-reverse, there’s no need either way. And in auto-reverse mechanisms, generally both capstans are always turning (in opposite directions), with only one pinch roller being depressed.
On the other hand, the reel motor is often controlled in wild and wacky ways (but its speed does not need to be controlled accurately at all). In most logic controlled cassette mechanisms, those motors not only wind the takeup reel at multiple speeds (for playback/recording and FF/rewind, often at multiple speeds to be gentle on the end of the tape, plus an extremely gentle low speed just to take up slack to prevent tangles before ejecting), but use pulsed high speeds in both directions to use centrifugal force to change directions, operate cams, etc. that (i.e. engage things that don’t at lower speeds), but also to overcome the friction in clutches and slip rings (i.e. to escape things that engage at lower speeds). This drives all the movements of the heads (including rotating the heads on auto-reverse units) and pinch rollers, reversing the reel direction, etc. (High-end decks use more motors and solenoids to make them faster.)
In Walkman type portable players, they often use one single motor to do all of this. It’s absolutely fascinating that one motor can literally do it all. The downside is that it makes them rather slow, and you hear the various clicky sounds as they sequence the motions to reverse the playback direction at the end of the tape, or go from FF/rewind to playback, for example. (Attached is the relevant page from the service manual of my Sony WM-EX508 Walkman, which I’ve owned since around 1997, which vaguely explains how the mechanism works.)