In this paper, we propose a new way to measure and test the presence of time-varying volatility in a discretely sampled jump-diffusion process that is contaminated by microstructure noise. We use the concept of pre-averaged truncated bipower variation to construct our t-statistic, which diverges in the presence of a heteroscedastic volatility term (and has a standard normal distribution otherwise). The test is inspected in a general Monte Carlo simulation setting, where we note that in finite samples the asymptotic theory is severely distorted by infinite-activity price jumps. To improve inference, we suggest a bootstrap approach to test the null of homoscedasticity. We prove the first-order validity of this procedure, while in simulations the bootstrap leads to almost correctly sized tests. As an illustration, we apply the bootstrapped version of our t-statistic to a large cross-section of equity high-frequency data. We document the importance of jump-robustness, when measuring heteroscedasticity in practice. We also find that a large fraction of variation in intraday volatility is accounted for by seasonality. This suggests that, once we control for jumps and deflate asset returns by a non-parametric estimate of the conventional U-shaped diurnality profile, the variance of the rescaled return series is often close to constant within the day.