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HEADis following a merge commit, thenHEAD^means the first parent ofHEAD, andHEAD^^(orHEAD^2) means the second parent ofHEAD. In a merge commit, the second parent ofHEADis not the same thing as the parent of the parent ofHEAD. See Ancestry References in the Git manual.HEAD^2does indeed indicate the second parent ofHEAD,HEAD^2andHEAD^^are not synonymous. Multiple carets do refer to commits in relation to the first parent, so just asHEAD^^refers in general to the grandparent ofHEAD,HEAD^^the grandparent of parent of the first commit ofHEADwhenHEADis a merge commit; likewise,HEAD^2^is the parent of the second parent.HEAD~~andHEAD~2are synonymous, so by analogy, I expected the number operator to work similarly with the caret. From what you are saying, it does not. That is unfortunate.