In this work, we present interpGaze, a novel framework for controllable gaze redirection that achieves both precise redirection and continuous interpolation. Given two gaze images with different attributes, our goal is to redirect the eye gaze of one person into any gaze direction depicted in the reference image or to generate continuous intermediate results. To accomplish this, we design a model including three cooperative components: an encoder, a controller and a decoder. The encoder maps images into a well-disentangled and hierarchically-organized latent space. The controller adjusts the magnitudes of latent vectors to the desired strength of corresponding attributes by altering a control vector. The decoder converts the desired representations from the attribute space to the image space. To facilitate covering the full space of gaze directions, we introduce a high-quality gaze image dataset with a large range of directions, which also benefits researchers in related areas. Extensive experimental validation and comparisons to several baseline methods show that the proposed interpGaze outperforms state-of-the-art methods in terms of image quality and redirection precision.