  Evil-avoiding permutations, introduced by Kim and Williams in 2022, arise in
the study of the inhomogeneous totally asymmetric simple exclusion process.
Rectangular permutations, introduced by Chiriv\`i, Fang, and Fourier in 2021,
arise in the study of Schubert varieties and Demazure modules. Taking a
suggestion of Kim and Williams, we supply an explicit bijection between
evil-avoiding and rectangular permutations in $S_n$ that preserves the number
of recoils. We encode these classes of permutations as regular languages and
construct a length-preserving bijection between words in these regular
languages. We extend the bijection to another Wilf-equivalent class of
permutations, namely the $1$-almost-increasing permutations, and exhibit a
bijection between rectangular permutations and walks of length $2n-2$ in a path
of seven vertices starting and ending at the middle vertex.
