SAGBI bases for rings of invariant Laurent polynomials

Let $k$ be a field, let $L_n = k[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$ be the Laurent polynomial ring in $n$ variables and let $G$ be a finite group of $k$-algebra automorphisms of $L_n$. We give a necessary and sufficient condition for the ring of invariants $L_n^G$ to have a SAGBI basis. We show that if this condition is satisfied, then $L_n^G$ has a SAGBI basis relative to any choice of coordinates in $L_n$ and any term order.