  Given a matching $M$ in the hypercube $Q^n$, the \emph{profile} of $M$ is the
vector $\boldsymbol{x}=(x_1,\ldots, x_n) \in \mathbb{N}^n$ such that $M$
contains $x_i$ edges whose endpoints differ in the $i$th coordinate. If $M$ is
a perfect matching, then it is clear that $||\boldsymbol{x}||_1 = 2^{n-1}$ and
it is easy to show that each $x_i$ must be even. Verifying a special case of a
conjecture of Balister, Gy\H{o}ri, and Schelp, we show that these conditions
are also sufficient.
