Through the application of Diagonal BEKK and Asymmetric Diagonal BEKK methodologies to intra-day data for eight cryptocurrencies, this paper investigates not only conditional volatility dynamics of major cryptocurrencies, but also their volatility co-movements. We first provide evidence that all conditional variances are significantly affected by both previous squared errors and past conditional volatility. It is also shown that both methodologies indicate that cryptocurrency investors pay the most attention to news relating to Neo and the least attention to news relating to Dash, while shocks in OmiseGo persist the least and shocks in Bitcoin persist the most, although all of the considered cryptocurrencies possess high levels of persistence of volatility over time. We also demonstrate that the conditional covariances are significantly affected by both cross-products of past error terms and past conditional covariances, suggesting strong interdependencies between cryptocurrencies. It is also demonstrated that the Asymmetric Diagonal BEKK model is a superior choice of methodology, with our results suggesting significant asymmetric effects of positive and negative shocks in the conditional volatility of the price returns of all of our investigated cryptocurrencies, while the conditional covariances capture asymmetric effects of good and bad news accordingly. Finally, it is shown that time-varying conditional correlations exist, with our selected cryptocurrencies being strongly positively correlated, further highlighting interdependencies within cryptocurrency markets.