Under what conditions is K-means clustering transformation-invariant?
The answer depends on your K-means algorithm, but what follows should work for standard algorithms.

You will get the same result if your transformation $T$ satisfies two conditions:


It preserves distances: $d(z,w) = d(T(z),T(w))$, where $d$ is your metric, say $d(z,w) = \|z-w\|$.
It preserves averages: if $\sum_i p_i z_i$ is a convex combination that $T(\sum_i p_i z_i) = \sum_i p_i T(z_i)$.


You can check this by going over the algorithm, showing that it always makes the same choices.