It is well known that Byzantine fault tolerant (BFT) consensus cannot be solved in the classic asynchronous message passing model when one-third or more of the processes may be faulty. Since many modern applications require higher fault tolerance, this bound has been circumvented by introducing non-equivocation mechanisms that prevent Byzantine processes from sending conflicting messages to other processes. The use of trusted hardware is a way to implement non-equivocation. Several different trusted hardware modules have been considered in the literature. In this paper, we study whether all trusted hardware modules are equivalent in the power that they provide to a system. We show that while they do all prevent equivocation, we can partition trusted hardware modules into two different power classes; those that employ shared memory primitives, and those that do not. We separate these classes using a new notion we call unidirectionality, which describes a useful guarantee on the ability of processes to prevent network partitions. We show that shared-memory based hardware primitives provide unidirectionality, while others do not.