As a player myself, and having seen much higher level player than me, reading the spin from the ball rotation (and in fact trajectory) of the ball is a common (if advanced) skill. Sometimes the movement of the bat can be deceptive (since with the same movement, where it contact on the bat, the finger pressure can affect the spin).

For example, backspin/underspin balls will move slower after the first bounce and feel 'damper' while topspin will jump. So it's def. possible (and in fact reliable) to read the spin from the spin and trajectory of the ball.