For an integer m >= 2, let Pm be the partition of the unit interval I into m equal subintervals, and let F-m be the class of piecewise linear maps on I with constant slope +/- m on each element of P-m. We investigate the effect on mixing properties when f is an element of F-m is composed with the interval exchange map given by a permutation sigma is an element of S-N interchanging the N subintervals of P-N. This extends the work in a previous paper [N. P. Byott, M. Holland and Y. Zhang, DCDS 33 (2013) 3365-3390], where we considered only the "stretch-and-fold" map f(sf) (x) = mx mod 1.