This study explores the design space of two-handed input (i.e., clicking or tapping with the thumb) on the touchpads of controllers for virtual reality (VR) interaction. Four experiments were conducted to fulfill this purpose. Experiment 1 investigated how users employed two VR controllers to perform four representative interaction tasks in VR and identified 14 potentially usable two-handed operations that involved tapping or clicking. Experiments 2 and 3 analyzed user performance of the 14 operations, providing insights into their interaction characteristics in terms of completion time, accuracy, and subjective feedback. In Experiment 4, we designed a command-input technique based on the proposed operations. We verified its effectiveness compared to context menus and marking menus in a VR text entry scenario. Our technique generally had shorter times and similar accuracy to the two menu types. Our work contributes to the design of VR interactions using two-handed controllers.