Summary: We study the following independent set reconfiguration problem, called TAR-Reachability: given two independent sets \(I\) and \(J\) of a graph \(G\), both of size at least \(k\), is it possible to transform \(I\) into \(J\) by adding and removing vertices one-by-one, while maintaining an independent set of size at least \(k\) throughout? This problem is known to be PSPACE-hard in general. For the case that \(G\) is a cograph on \(n\) vertices, we show that it can be solved in time \(O(n^2)\), and that the length of a shortest reconfiguration sequence from \(I\) to \(J\) is bounded by \(4n-2k\) (if it exists). More generally, we show that if \(\mathcal G\) is a graph class for which (i) TAR-Reachability can be solved efficiently, (ii) maximum independent sets can be computed efficiently, and which satisfies a certain additional property, then the problem can be solved efficiently for any graph that can be obtained from a collection of graphs in \(\mathcal G\) using disjoint union and complete join operations. Chordal graphs and claw-free graphs are given as examples of such a class \(\mathcal G\).