In the past decade, deep learning-based networks have achieved unprecedented success in numerous tasks, including image classification. Despite this remarkable achievement, recent studies have demonstrated that such networks are easily fooled by small malicious perturbations, also known as adversarial examples. This security weakness led to extensive research aimed at obtaining robust models. Beyond the clear robustness benefits of such models, it was also observed that their gradients with respect to the input align with human perception. Several works have identified Perceptually Aligned Gradients (PAG) as a byproduct of robust training, but none have considered it as a standalone phenomenon nor studied its own implications.In this work, we focus on this trait and test whether Perceptually Aligned Gradients imply Robustness. To this end, we develop a novel objective to directly promote PAG in training classifiers and examine whether models with such gradients are more robust to adversarial attacks. Extensive experiments on CIFAR-10 and STL validate that such models have improved robust performance, exposing the surprising bidirectional connection between PAG and robustness.