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In recent years, convolutional neural network has shown good performance in many image processing and computer vision tasks. However, a standard CNN model is not invariant to image rotations. In fact, even slight rotation of an input image will seriously degrade its performance. This shortcoming precludes the use of CNN in some practical scenarios. Thus, in this paper, we focus on designing convolutional layer with good rotation invariance. Specifically, based on a simple rotation-invariant coordinate system, we propose a new convolutional operation, called Rotation-Invariant Coordinate Convolution (RIC-C). Without additional trainable parameters and data augmentation, RIC-C is naturally invariant to arbitrary rotations around the input center. Furthermore, we find the connection between RIC-C and deformable convolution, and propose a simple but efficient approach to implement RIC-C using Pytorch. By replacing all standard convolutional layers in a CNN with the corresponding RIC-C, a RIC-CNN can be derived. Using MNIST dataset, we first evaluate the rotation invariance of RIC-CNN and compare its performance with most of existing rotation-invariant CNN models. It can be observed that RIC-CNN achieves the state-of-the-art classification on the rotated test dataset of MNIST. Then, we deploy RIC-C to VGG, ResNet and DenseNet, and conduct the classification experiments on two real image datasets. Also, a shallow CNN and the corresponding RIC-CNN are trained to extract image patch descriptors, and we compare their performance in patch verification. These experimental results again show that RIC-C can be easily used as drop in replacement for standard convolutions, and greatly enhances the rotation invariance of CNN models designed for different applications.