Letting (t n) denote the Thue-Morse sequence with values 0, 1, we note that the Woods-Robbins product n≥0 2n + 1 2n + 2 (−1) tn = 2 −1/2 involves a rational function in n and the ±1 Thue-Morse sequence ((−1) tn) n≥0. The purpose of this paper is twofold. On the one hand, we try to find other rational functions for which similar infinite products involving the ±1 Thue-Morse sequence have an expression in terms of known constants. On the other hand, we also try to find (possibly different) rational functions R for which the infinite product R(n) tn also has an expression in terms of known constants.