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Robustness to transformation is desirable in many computer vision tasks, given that input data often exhibits pose variance. While translation invariance and equivariance is a documented phenomenon of CNNs, sensitivity to other transformations is typically encouraged through data augmentation. We investigate the modulation of complex valued convolutional weights with learned Gabor filters to enable orientation robustness. The resulting network can generate orientation dependent features free of interpolation with a single set of learnable rotation-governing parameters. By choosing to either retain or pool orientation channels, the choice of equivariance versus invariance can be directly controlled. Moreover, we introduce rotational weight-tying through a proposed cyclic Gabor convolution, further enabling generalisation over rotations. We combine these innovations into Learnable Gabor Convolutional Networks (LGCNs), that are parameter-efficient and offer increased model complexity. We demonstrate their rotation invariance and equivariance on MNIST, BSD and a dataset of simulated and real astronomical images of Galactic cirri.