How about mirror symmetry of Calabi-Yaus? This started from the observation by physicists that string theory on certain pairs of Calabi-Yaus gave identical theories. This has lead to a lot of work by physicists and mathematicians to understand what's going on, leading to things like the SYZ conjecture, homological mirror symmetry, etc.

So, more specifically physicists theories treat spacetime $M$ as something that locally looks like $M=\mathbb{R}^4\times X$ in such a way that $X$ is "small" by saying (roughly) operators (which represent observables) when "looking at things" below a certain energy scale can't see directly the dynamics associated with $X$. Associated with $M$ is a special kind of quantum field theory called a superconformal field theory (SCFT), which requires that $X$ be a Calabi-Yau 3-fold.

Various topological invariants of $X$ can tell us about how the SCFT behaves.

But it was discovered that the associated SCFTs don't uniquely determine $X$. It turns out there are pairs of Calabi-Yau 3-folds $(X,\hat{X})$ (called mirrors) that give the same SCFT.

From the SCFT point of view, these two mirror manifolds are related by an automorphism of the SCFT, which does not correspond to an automorphism of the Calabi-Yau manifold, but instead gives a mirror manifold in a way that switches cohomology groups around. It can also be thought of as switching complex structures with symplectic ones somehow.

From the rigorous point of view, though, not much of this is well-defined. It relies on the machinery of QFTs which no one has been able to come close to defining axiomatically, as well as string theory which relies on a lot of machinery that has the same kinds of problems.

Out of this came a number of more mathematically precise conjectures, such as the SYZ conjecture, which explains this in terms of special Lagrangian manifolds and fibrations of the mirror manifolds into it.

This also started ideas of homological mirror symmetry, which tries to describe this in terms of homology and derived categories.