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We show that interlayer spins play a dual role in the Cairo pentagonal magnetBi$_4$Fe$_5$O$_{13}$F, on one hand mediating the three-dimensional (3D)magnetic order and on the other driving spin-reorientation transitions bothwithin and between the planes. The corresponding sequence of magnetic ordersunraveled by neutron diffraction and M\"ossbauer spectroscopy features twoorthogonal magnetic structures described by opposite local vector chiralities,and an intermediate, partly disordered phase with nearly collinear spins. Asimilar collinear phase has been predicted theoretically to be stabilized byquantum fluctuations, but Bi$_4$Fe$_5$O$_{13}$F is very far from the relevantparameter regime. While the observed in-plane reorientation cannot be explainedby any standard frustration mechanism, our ab initio band-structurecalculations reveal strong single-ion anisotropy of the interlayer Fe$^{3+}$spins that turns out to be instrumental in controlling the local vectorchirality and the associated interlayer order.

Collinear order and chirality-reorientation transition in the Cairo pentagonal magnet Bi$_4$Fe$_5$O$_{13}$F