  We investigate commutative images of languages recognised by register
automata and grammars. Semi-linear and rational sets can be naturally extended
to this setting by allowing for orbit-finite unions instead of only finite
ones. We prove that commutative images of languages of one-register automata
are not always semi-linear, but they are always rational. We also lift the
latter result to grammars: commutative images of one-register context-free
languages are rational, and in consequence commutatively equivalent to register
automata. We conjecture analogous results for automata and grammars with
arbitrarily many registers.
