  The main conjecture of Iwasawa theory is a conjecture on the relation between
a Selmer group and a conjectural $p$-adic $L$-function. This conjectural
$p$-adic $L$-function is expected to satisfy a conjectural functional equation
in a certain sense. In view of the main conjecture and this conjectural
functional equation, one would expect to have certain algebraic relationship
between the Selmer group attached to a Galois representation and the Selmer
group attached to the Tate twist of the dual of the Galois representation. It
is precisely a component of this algebraic relationship that this paper aims to
investigate. Namely, for a given "ordinary" $p$-adic representation, we compare
its Selmer group with the Selmer group of its Tate dual over an admissible
$p$-adic Lie extension, and show that the generalized Iwasawa $\mu$-invariants
associated to the Pontryagin dual of the two said Selmer groups are the same.
We should mention that in proving the said equality of the $\mu$-invariants, we
do not assume the main conjecture nor the conjectural functional equation.
