We investigate the computing power of stateless multicounter machines with reversal-bounded counters. Such a machine has m-counters operating on a one-way input delimited by left and right end markers. The move of the machine depends only on the symbol under the input head and the signs of the counters (zero or positive). At each step, every counter can be incremented by 1, decremented by 1 (if it is positive), or left unchanged. An input string is accepted if, when the input head is started on the left end marker with all counters zero, the machine eventually reaches the configuration where the input head is on the right end marker with all the counters again zero. Moreover, for a specified k, no counter makes more than k-reversals (i.e., alternations between increasing mode and decreasing mode and vice-versa) on any computation, accepting or not. We mainly focus our attention on deterministic realtime (the input head moves right at every step) machines. We show hierarchies of computing power with respect to the number of counters and reversals. It turns out that the analysis of these machines gives rise to rather interesting combinatorial questions.