On definable Galois groups and the strong canonical base property

In \cite{HPP}, Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that $T$ has the canonical base property in a strong form, " internality to" being replaced by "algebraicity in". In the current paper we give a reasonably robust definition of the "strong canonical base property" in a rather more general finite rank context than \cite{HPP}, and prove its {\em equivalence} with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that $1$-based groups are rigid.