  We consider the problem of finding a bijection between the sets of
alternating sign matrices and of totally symmetric self complementary plane
partitions, which can be reformulated using Gog and Magog triangles. In a
previous work we introduced GOGAm triangles, which are images of Magog
triangles by the Sch\"utzenberger involution. In this paper we introduce left
Gog and GOGAm trapezoids. We conjecture that they are equienumerated, and we
give an explicit bijection between such trapezoids with one or two diagonals.
We also study the distribution of inversions and coinversions in Gog triangles.
