First of all, usually implementations are centered around the revised dual simplex, not the primal (even though solvers will still use a primal simplex method implementation for some tasks in the solution process).
According to Huangfu and Hall and Koberstein, the most important non-textbook techniques for the dual revised simplex appear to be:
Dual Steepest Edge algorithm for choosing the variable leaving the basis Forrest, Goldfarb: Steepest-edge simplex algorithms for linear programming
Bound Flipping Ratio Test See for example Koberstein
It is based on the observation that the reduced cost value of a boxed non-basic primal variable can be kept dual feasible even if it switches sign by setting the variable it to its opposite bound. This means that the dual step length can be further increased and breakpoints associated with boxed primal variables can be skipped as long as the dual objective function keeps improving
Using LU factorization No solver I know actually inverts the basis matrix to solve the linear systems. Simplex implementations typically compute a (sparse)LU factorization and update it using Forrest-Tomlin updates to avoid too-frequent recomputation of the factorization.
Exploiting Hypersparsity Exploiting the fact that when solving a linear system the right hand side is soemtimes/often sparse. See for example Hall, McKinnon
Exploiting parallelism While the linear algebra in a sparse simplex implementation is not trivial to parallelize, there are some things that can be done to make use of parallel machines. See for example Huangfu, Hall and Hall.
Note that much of the speedup of solvers compared to one another is often due to a better presolve and/or crashstart implementation. I suspect some of the better open-source simplex implementations to not perform much worse than commercial implementations.