  Alternating sign matrices and totally symmetric self-complementary plane
partitions are equinumerous sets of objects for which no explicit bijection is
known. In this paper, we identify a subset of totally symmetric
self-complementary plane partitions corresponding to permutations by giving a
statistic-preserving bijection to permutation matrices, which are a subset of
alternating sign matrices. We use this bijection to define a new partial order
on permutations, and prove this new poset contains both the Tamari lattice and
the Catalan distributive lattice as subposets. We also study a new partial
order on totally symmetric self-complementary plane partitions arising from
this perspective and show that this is a distributive lattice related to Bruhat
order when restricted to permutations.
