Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while preserving computability, I will introduce invariant cup and Massey products — and, more generally, an A-infinity structure — on the linearized LCH. I will apply the products and A-infinity structure to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors (and maybe a few more applications). This is joint work with G. Civan, J. Etnyre, P. Koprowski, and A. Walker.