This tip is a bit technical mostly for higher-level players, but it is also for those who wish to understand the game at those higher levels. Thinking about spins and reacting to them in the way that we'll here might give you a headache. You have been warned.
With practice, most players can learn to push short against a simple short backspin serve. (Here's a Tip of the Week, Pushing Short.) However, it takes lots of practice to be able to do this in a match where you don't know in advance what type of serve you'll be getting. The problem with pushing short isn't so much against backspin serves, but against sidespin serves, which are often pushed back high and to the side. (Another problem is pushing short against no-spin serves that look like backspin, where you have to chop down on the ball, but that's a separate issue.) There's a somewhat simple trick for pushing these sidespin serves short.
Imagine a forehand pendulum serve (or another serve with this type spin) coming short to your backhand with mostly sidespin and perhaps some backspin. (We're assuming both players are righties; lefties can adjust.) If you try pushing short and don't adjust properly, the ball pops up and to your right. It's almost like pushing against a topspin. Now drop your racket tip some, and meet the ball a little on the bottom right side with a grazing motion, with a slightly downward motion. Instead of popping up to your right as it would when you push against sidespin, it'll go out as if you were pushing a backspin because you are now meeting the spin head-on, as if pushing against a backspin. With this technique, you'll find it much easier to drop the ball short to the left, i.e. short to the opponent's forehand.
Now imagine a backhand serve (or another serve with this type spin) coming short to your backhand with mostly sidespin and perhaps some backspin. If you try pushing short and don't adjust properly, the ball pops up and to your left. Again, it's almost like pushing against a topspin. Now raise your racket tip some, and meet the ball a little on the bottom left side with a grazing motion, with a slightly downward motion. Instead of popping up to the left as it would when you push against sidespin, it'll go out as if you were pushing a backspin because you are again meeting the spin head-on, as if pushing against a backspin. With this technique, you'll find it much easier to drop the ball short to the right, i.e. short to the opponent's backhand.
With some adjustments, you can use this type of technique to drop the ball anywhere, but in general on the backhand side it's easier to drop a forehand pendulum serve type spin short to the forehand, and a backhand serve type spin to the backhand side. You can make the same adjustments with your forehand push, raising the racket tip some against forehand pendulum serve type spins, lowering it against backhand serve type spins. With practice, you'll find dropping the ball short against a short sidespin to be as easy as doing it against a short backspin. (Of course, pushing short against short backspin isn't "easy" unless you practice it to develop the light touch necessary - but it's a valuable tool to have against many players, so learn to do so.)
If you are having trouble visualizing all this, then imagine the opponent's forehand pendulum sidespin serve. Take a ball and actually rotate it with that type of spin as it comes toward you. Now imagine pushing it, and see how it would jump to the right (the opponent's backhand). Now drop the racket tip, and presto - you are meeting the spin head-on, like pushing backspin to backspin, and thereby getting much more control, just as you do when pushing against a backspin.
The same technique can also be used to push long, but is not always as effective there since for deep pushes you want to maximize the backspin. For this, you'd more likely just aim the opposite way as the sidespin, and chop down on the ball.