We will explain a general strategy for studying type II$_1$ factors, which consists in ``playing amenability against rigidity'' whenever some (weak) versions of these properties are met. The coexistence of these oposing properties create enough ``tension'' within the algebra to unfold much of its structure. We will exemplify with three types of applications and results: 1). When the II$_1$ factor $M$ contains Cartan subalgebras $A\subset M$ such that $A\subset M$ satisfies a combination of ``aT-menability'' and ``(T)'' properties. 2). When $M=R \rtimes_\sigma G$ with $\sigma$ a ``malleable action'' (e.g., Bernoulli shift) and $G$ contains an infinite normal relatively rigid subgroup. 3). When $M$ is a tensor product of factors satisfying Ozawa's property (AO).