Let M be a matroid. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. This result was extended by Oxley, Semple, and Whittle, who showed that, when M is 3-connected, there is a corresponding tree decomposition that dis- plays all non-trivial 3-separations of M up to a certain natural equivalence. This equivalence is based on the notion of the full closure fcl(Y ) of a set Y in M, which is obtained by beginning with Y and alternately applying the closure op- erators of M and M* until no new elements can be added. Two 3-separations (Y₁,Y₂) and (Z₁,Z₂) are equivalent if {fcl(Y₁), fcl(Y₂)} = {fcl(Z₁), fcl(Z₂)}. The purpose of this paper is to identify all the structures in M that lead to two 3-separations being equivalent and to describe the precise role these structures have in determining this equivalence.