Characterization of substitution invariant 3iet words

We study infinite words coding an orbit under an exchange of three intervals which have full complexity $\text{C}(n)=2n+1$ for all $n\in \mathbb{N}$ (non-degenerate 3iet words). In terms of parameters of the interval exchange and the starting point of the orbit we characterize those 3iet words which are invariant under a primitive substitution. Thus, we generalize the result recently obtained for sturmian words.