Extracting geometric features from 3D point clouds is widely applied in many tasks, including registration and recognition. We propose a simple yet effective method, termed height-azimuth image based transformation-invariant net (HA-TiNet), to learn a distinctive, general and rotation-invariant 3D local descriptor. HA-TiNet is composed of a height-azimuth image generator and a feature extraction net. Based on a local reference axis (LRA), the height-azimuth image generator first partitions local region along the plane-radial direction, and then implements a statistic of height and azimuth information in each divided space to generate a set of height-azimuth images. The generated height-azimuth images are invariant in the rotation around x- and y-axes and have high accuracy due to the high repeatability of an LRA. Besides, they can be easily embedded in 2D convolutional neural networks (CNNs). Our feature extraction net learns the information on the height-azimuth images using a ResNet-based backbone and a rotation-invariant layer. The ResNet-based backbone is lightweight while very effective. The rotation-invariant layer removes the rotation-variance around z-axis, making our descriptor have full rotation-invariance. Extensive experiments on indoor and outdoor datasets show that our method presents superior overall performance, and exhibits strong descriptiveness and generalization ability compared to the state-of-the-art descriptors.