Program-specific or function-specific optimization phase sequences are universally accepted to achieve better overall performance than any fixed optimization phase ordering. A number of heuristic phase order space search algorithms have been devised to find customized phase orderings achieving high performance for each function. However, to make this approach of iterative compilation more widely accepted and deployed in mainstream compilers, it is essential to modify existing algorithms, or develop new ones that find near-optimal solutions quickly. As a step in this direction, in this paper we attempt to identify and understand the important properties of some commonly employed heuristic search methods by using information collected during an exhaustive exploration of the phase order search space. We compare the performance obtained by each algorithm with all others, as well as with the optimal phase ordering performance. Finally, we show how we can use the features of the phase order space to improve existing algorithms as well as devise new, and better performing search algorithms.