(Credit goes to @EdgarFigaro who found that glitch)

By using momentum, you can clip into any wall in the game for one frame. "Walls" in this case include actual walls, but also the side of 1-block tall platforms, like moving platforms in the blue temple.

The steps for executing the glitch: (assuming you want to jump on a right wall, but it works on left walls as well)

• Run towards the wall, so you will build momentum
• Jump from the ground before reaching the wall
• Slightly before reaching the wall, quick-tap left to change direction. The momentum from the run will still carry you to the right, towards the wall. If you press left for too long, you will either stop in place or start moving left, and then you won't touch the wall.
• When you reach the wall (with the back of your character touching it), press right to turn your character. That turn will clip your character one pixel inside the wall for a few milliseconds. You know it worked if your character has stopped falling for a moment, and hear the landing sound effect.
• Immediately after clipping inside the wall, press up and you will be able to jump from the wall. The timing between turning and jumping is precise.

It might be possible to chain several wall jumps one after the other, but because you need to build enough momentum to reach the wall each time (while looking on the other direction); it appears to be very hard or impossible for humans to execute.

Here is a show case of the glitch: