We begin by introducing an interesting class of functions, known as the Schemmel totient functions, that generalizes the Euler totient function. For each Schemmel totient function L_m, we define two new functions, denoted R_m and H_m, that arise from iterating L_m. Roughly speaking, R_m counts the number of iterations of L_m needed to reach either 0 or 1, and H_m takes the value (either 0 or 1) that the iteration trajectory eventually reaches. Our first major result is a proof that, for any positive integer m, the function H_m is completely multiplicative. We then introduce an iterate summatory function, denoted D_m, and define the terms D_m-deficient, D_m-perfect, and D_m-abundant. We proceed to prove several results related to these definitions, culminating in a proof that, for all positive even integers m, there are infinitely many D_m-abundant numbers. Many open problems arise from the introduction of these functions and terms, and we mention a few of them, as well as some numerical results.