  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, we introduce invariant
cup and Massey products - and, more generally, an A_\infty structure - on the
linearized LCH. We apply the products and A_\infty structure in three ways: to
find infinite families of Legendrian knots that are not isotopic to their
Legendrian mirrors, to reinterpret the duality theorem of the fourth author in
terms of the cup product, and to recover higher-order linearizations of the
LCH.
