Cosmic polarization rotation, which may result from parity-violating new physics or the presence of primordial magnetic fields, converts $E$-mode polarization of the cosmic microwave background (CMB) into $B$-mode polarization. Anisotropic cosmic polarization rotation leads to statistical anisotropy in CMB polarization and can be reconstructed with quadratic estimator techniques similar to those designed for gravitational lensing of the CMB. At the sensitivity of upcoming CMB surveys, lensing-induced $B$-mode polarization will act as a limiting factor in the search for anisotropic cosmic polarization rotation, meaning that an analysis which incorporates some form of delensing will be required to improve constraints on the effect with future surveys. In this paper we extend the ResUNet-CMB convolutional neural network to reconstruct anisotropic cosmic polarization rotation in the presence of gravitational lensing and patchy reionization, and we show that the network simultaneously reconstructs all three effects with variance that is lower than that from the standard quadratic estimator nearly matching the performance of an iterative reconstruction method.