The thing to remember here is that what is traditionally described as "leg spin" and "off spin" is actually a description of the technique used to spin the ball: leg spinners use the wrist to impart spin, while off spinners use the fingers. They are completely different techniques, which is why bowlers typically do only one or the other.
That said, both wrist spinners and finger spinners have always had variations which do spin differently from the their stock ball. While for a right-armed wrist spinner the stock ball turns from right to left as viewed by the bowler:
- The googly turns from left to right.
- The topspinner has no significant turn but instead has extra bounce, aiming to induce an error from a batter who expects the ball to turn when it does not.
- The flipper is the inverse of the topspinner and has no significant turn but instead less bounce.
There are then also balls with mixed spin, for example the slider which is a combination of the stock ball's sidespin and the flipper's backspin. As you can see from all those, wrist spinners already have the ability to spin the ball so that the axis of rotation is pointing in almost any direction1.
Finger spinners have traditionally had less variation; the arm ball was for many years essentially the only variation, again achieving success due to the lack of turn rather than turning significantly in the opposite direction. In recent years, the doosra, which does genuinely turn in the opposite direction to the finger spinner's stock ball, has been invented, along with controversy and changes to the Laws as to whether it can be bowled legally or not. Finger spinners have also been experimenting with other techniques, most notably the carrom ball.
tl;dr: spinners do spin the ball in "both" directions, they just do it via different techniques.
1. Any direction with the axis of rotation parallel to the ground anyway - i.e. pitch and roll in aeronautical terms. No spinner can produce any significant amount of "yaw" spin with the axis of rotation perpendicular to the ground just because of the way the arm works.