We construct a class of rank-one infinite measure-preserving transformations such that for each transformation T in the class, the cartesian product T × T of the transformation with itself is ergodic, but the product T ×T −1 of the transformation with its inverse is not ergodic. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.