  We show that, given some lacunary sequence of angles
$\mathbf{\theta}=(\theta_j)_{j\in\N}$ not converging too fast to zero, it is
possible to build a rare differentiation basis $\mathcal{B}$ of rectangles
parallel to the axes that differentiates $L^1(\mathbb{R}^2)$ while the basis
$\mathcal{B}_{\mathbf{\theta}}$ obtained from $\mathcal{B}$ by allowing its
elements to rotate around their lower left vertex by the angles $\theta_j$,
$j\in\mathbb{N}$, fails to differentiate all Orlicz spaces lying between
$L^1(\mathbb{R}^2)$ and $L\log L(\mathbb{R}^2)$.
