A square matrix is of maximal trace over rotations exactly when it is symmetric and has at most one negative eigenvalue whose absolute value does not exceed the others.
and Witzgall C., A Purely Algebraic Justification of the Kabsch-Umeyama Algorithm, Journal of Research of the National Institute of Standards and Technology 124 (2019) 1-6
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Characterization and Computation of Matrices of Maximal Trace over Rotations
A square matrix is of maximal trace over rotations exactly when it is symmetric and has at most one negative eigenvalue whose absolute value does not exceed the others.