Summary: Let \(\mathcal{O}\) be a set of \(k\) orientations in the plane, and let \(P\) be a simple polygon in the plane. Given two points \(p, q\) inside \(P\), we say that \(p\mathcal{O}\)-sees \(q\) if there is an \(\mathcal{O}\)-staircase contained in \(P\) that connects \(p\) and \(q\). The \(\mathcal{O}\)-Kernel of the polygon \(P\), denoted by \(\mathcal{O}\)-Kernel\((P)\), is the subset of points of \(P\) which \(\mathcal{O}\)-see all the other points in \(P\). This work initiates the study of the computation and maintenance of \(\mathcal{O}\)-Kernel\((P)\) as we rotate the set \(\mathcal{O}\) by an angle \(\theta\), denoted by \(\mathcal{O}\)-Kernel\(_{\theta}(P)\). In particular, we consider the case when the set \(\mathcal{O}\) is formed by either one or two orthogonal orientations, \( \mathcal{O}=\{0^\circ\}\) or \(\mathcal{O}=\{0^\circ ,90^\circ\}\). For these cases and \(P\) being a simple polygon, we design efficient algorithms for computing the \(\mathcal{O}\)-Kernel\(_{\theta}(P)\) while \(\theta\) varies in \([-\frac{\pi}{2},\frac{\pi}{2})\), obtaining: (i) the intervals of angle \(\theta\) where \(\mathcal{O}\)-Kernel\(_{\theta}(P)\) is not empty, (ii) a value of angle \(\theta\) where \(\mathcal{O}\)-Kernel\(_{\theta }(P)\) optimizes area or perimeter. Further, we show how the algorithms can be improved when \(P\) is a simple orthogonal polygon. In addition, our results are extended to the case of a set \(\mathcal{O}=\{\alpha_1,\dots,\alpha_k\}\).