  We define a bi-skew brace to be a set $G$ with two group operations $\star$
and $\circ$ so that $(G, \circ, \star)$ is a skew brace with additive group
$(G, \star)$ and also with additive group $(G, \circ)$. If $G$ is a skew brace,
then $G$ corresponds to a Hopf Galois structure of type $(G, \star)$ on any
Galois extension of fields with Galois group isomorphic to $(G, \circ)$. If $G$
is a bi-skew brace, then $G$ also corresponds to a Hopf Galois structure of
type $(G, \circ)$ on a Galois extension of fields with Galois group isomorphic
to $(G, \star)$. Many non-trivial examples exist. One source is radical rings
$A$ with $A^3 = 0$, where one of the groups is abelian and the other need not
be. The left braces of degree $p^3$ classified by Bachiller are bi-skew braces
if and only they are radical rings. A different source of bi-skew braces is
semidirect products of arbitrary finite groups, which yield many examples where
both groups are non-abelian, and a skew brace proof of a result of Crespo, Rio
and Vela that if $G = H\rtimes J$ is a semidirect product of finite groups,
then a Galois extension of fields with Galois group $G$ has a Hopf Galois
structure of type $H \times J$.
