Truncated survival data arise when the failure time is observed only if it falls within a subject-specific truncating set. Most analysis methods rely on the key assumption of quasi-independence, that is, factorization of the joint density of failure and truncation times into a product proportional to the individual densities in the observable region. Unlike independence of failure time and censoring time, quasi-independence can be tested. Tests of quasi-independence are available for one-sided truncation and for truncation that depends on a measured covariate, but not for more complex truncation schemes. Here tests of quasi-independence based on a multivariate conditional Kendall's tau are proposed for doubly truncated data, bivariate left-truncated data, and other forms of truncated survival data that arise when initiating or terminating event times are interval-censored. Asymptotic properties under the null are derived. The tests are illustrated using several real datasets and evaluated via simulation.