A 4-dimensional Riemannian manifold $M$, equipped with an additional tensor structure $S$, whose fourth power is minus identity, is considered. The structure $S$ has a skew-circulant matrix with respect to some basis and $S$ acts as an isometry with respect to the metric $g$. A fundamental tensor is defined on such a manifold $(M,g,S)$ by $g$ and by the covariant derivative of $S$. This tensor satisfies a characteristic identity which is invariant to the usual conformal transformation. Some curvature properties of $(M,g,S)$ are obtained. A Lie group as a manifold of the considered type is constructed. A Hermitian manifold associated with $(M,g,S)$ is also considered. It turns out that it is a locally conformal Kähler manifold.