We present a newly validated and refined framework, TA-PIVOT (TA Practices In and Views Of Teaching), for examining how physics TAs talk about and how they engage in physics teaching. This work builds upon and extends prior efforts to characterize instructors' beliefs and practices by examining both domains in parallel. We present the comprehensive framework (developed from a study of 31 total TAs) and demonstrate its utility in analyzing both interviews and classroom video observations for a sample of eight TAs. We also discuss how this framework may be used to examine variation in beliefs and practices, track the development of beliefs over time, and inform TA preparation. [This paper is part of the Focused Collection on Preparing and Supporting University Physics Educators.]