  We define an infinite measure-preserving transformation to have infinite
symmetric ergodic index if all finite Cartesian products of the transformation
and its inverse are ergodic, and show that infinite symmetric ergodic index
does not imply that all products of powers are conservative, so does not imply
power weak mixing. We provide a sufficient condition for $k$-fold and infinite
symmetric ergodic index and use it to answer a question on the relationship
between product conservativity and product ergodicity. We also show that a
class of rank-one transformations that have infinite symmetric ergodic index
are not power weakly mixing, and precisely characterize a class of power weak
transformations that generalizes existing examples.
