We present Learning to Place by Picking (LPP), a method capable of autonomously collecting demonstrations for a family of placing tasks in which objects must be manipulated to specific locations. With LPP, we approach the learning of robotic object placement policies by reversing the grasping process and exploiting the inherent symmetry of the pick and place problems. Specifically, we obtain placing demonstrations from a set of grasp sequences of objects that are initially located at their target placement locations. Our system is capable of collecting hundreds of demonstrations without human intervention by using a combination of tactile sensing and compliant control for grasps. We train a policy directly from visual observations through behaviour cloning, using the autonomously-collected demonstrations. By doing so, the policy can generalize to object placement scenarios outside of the training environment without privileged information (e.g., placing a plate picked up from a table and not at the original placement location). We validate our approach on home robotic scenarios that include dishwasher loading and table setting. Our approach yields robotic placing policies that outperform policies trained with kinesthetic teaching, both in terms of performance and data efficiency, while requiring no human supervision.