  We show that $\phi$-invariant submanifolds of metric contact pairs with
orthogonal characteristic foliations make constant angles with the Reeb vector
fields. Our main result is that for the normal case such submanifolds of
dimension at least $2$ are all minimal. We prove that an odd-dimensional
$\phi$-invariant submanifold of a metric contact pair with orthogonal
characteristic foliations inherits a contact form with an almost contact metric
structure, and this induced structure is contact metric if and only if the
submanifold is tangent to one Reeb vector field and orthogonal to the other
one. Furthermore we show that the leaves of the two characteristic foliations
of the differentials of the contact pair are minimal. We also prove that when
one Reeb vector field is Killing and spans one characteristic foliation, the
metric contact pair is a product of a contact metric manifold with
$\mathbb{R}$.
