We introduce a class of generally applicable specification tests for constant and dynamic structures of conditional correlations in multivariate GARCH models. The tests are robust to the presence of time-varying higher-order conditional moments of unknown form and are pure significance tests. The tests can identify linear and nonlinear misspecifications in conditional correlations. Our approach does not necessitate a particular parameter estimation method and distributional assumption on the error process. The asymptotic distribution of the tests is invariant to the uncertainty in parameter estimation. We assess the finite sample performance of our tests using simulated and real data.