Comment by lathyrus_long

As others note, it's exactly a change in basis. In particular, it's an orthonormal basis, meaning that the dot product (correlation) between a basis function and itself is one, and between any two different basis functions is zero. This gives some intuition for why the forward and inverse transforms look so similar, in the same way that the inverse of an orthonormal matrix (like a rotation matrix) is just its transpose.