Darwiche and Pearl's seminal 1997 article outlined a number of baseline\nprinciples for a logic of iterated belief revision. These principles, the DP\npostulates, have been supplemented in a number of alternative ways. Most of the\nsuggestions made have resulted in a form of `reductionism' that identifies\nbelief states with orderings of worlds. However, this position has recently\nbeen criticised as being unacceptably strong. Other proposals, such as the\npopular principle (P), aka `Independence', characteristic of `admissible'\nrevision operators, remain commendably more modest. In this paper, we\nsupplement both the DP postulates and (P) with a number of novel conditions.\nWhile the DP postulates constrain the relation between a prior and a posterior\nconditional belief set, our new principles notably govern the relation between\ntwo posterior conditional belief sets obtained from a common prior by different\nrevisions. We show that operators from the resulting family, which subsumes\nboth lexicographic and restrained revision, can be represented as relating\nbelief states that are associated with a `proper ordinal interval' (POI)\nassignment, a structure more fine-grained than a simple ordering of worlds. We\nclose the paper by noting that these operators satisfy iterated versions of a\nlarge number of AGM era postulates, including Superexpansion, that are not\nsound for admissible operators in general.\n