The DFT shift theorem implies that any circular shift in the input space is equivalent to a phase change in the frequency domain, while the absolute values are preserved.
$$ \mathcal{F}(\{x_{n-m}\})_k=\mathcal{F}(\{x_n\})_k\cdot e^{-\frac{2\pi i}{N}k m} $$
A circular shift can be represented as a multiplication by a particular orthogonal matrix, and DFT is a special kind of unitary transformation.
I wonder if there are generalizations of the shift theorem to wider classes of input transformations than circular shifts and DFT, such that the original transformation always looks like a phase change in the new representation?