We call a linear code $C$ with length $n$ over a field $F$, a linear complementary equi-dual code, when there exists a linear code $D$ over $F$ such that $D$ is permutation equivalent to $C^\perp$ and $(C,D)$ is a linear complementary pair of codes, that is, $C+ D=F^n$ and $C\cap D=0$. We first state a necessary condition on a code $C$ to be linear complementary equi-dual. Then, we conjecture that this necessary condition is also sufficient and present several statements which support this conjecture.