Thank you for the comment, Ridha! I don’t believe that’s accurate, but want to share my thinking — please correct me if I’m wrong.

If X is a matrix with n rows (samples) and p columns (variables), then XX’ would be an n*n matrix, whereas X’X would be a p*p matrix. The covariance matrix that measures the covariance of each column should be p*p, since there are p variables.