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This paper addresses global asymptotic stability (GAS) and stabilization issues of switched linear time-invariant (LTI) systems with saddle points and time-varying switching delays. Firstly, two criteria of GAS are proposed for such switched systems with all subsystems sharing a common unique saddle point under arbitrary periodic/quasi-periodic switching paths (PSP/QSP). Secondly, by using our stability results we design global asymptotic-stabilizing controls (GASC), periodic/quasi-periodic stabilizing switching paths (PSSP/QSSP), and an algorithm of GASC and PSSP/QSSP for the switched systems. Finally, we present a numerical example of a switching multi-agent system to illustrate the effectiveness and practicality of our new results. [ABSTRACT FROM AUTHOR]