In this talk, I will present new symmetries of classical black hole mechanics which are relevant both at the perturbative and the non-perturbative levels. I will focus on simplified configuration which can be treated as a one dimensional field theory. A first part of the talk will be devoted to the problem of the vanishing of Love numbers (which encode the deformability of a self-gravitating object) in 4d black hole solutions of GR, a rather surprising feature compared to GR in higher dimensions or modified gravity. Several proposals exist and the origin of these symmetries is still debated. In this talk, I will discuss a specific set of symmetries (in relation to the one found by L. Hui and collaborators in arxiv:2105.01069) for the static black hole perturbations and explain how they can used to explain the vanishing of the Love numbers. In the second part of the talk, I will explain how the same type of symmetry can be realized at the non-perturbative level in black hole mechanics. Finally, I will explain how they can be used to improve the quantization of black hole mini-superspace and introduce non-linear extension to Wheeler-de Witt equation.