This paper develops two conditionally heteroscedastic models which allow an asymmetric reaction of the conditional volatility to the arrival of news. Such a reaction is induced by both the sign of past shocks and the size of past unexpected volatility. The proposed models are shown to converge in distribution to absolutely continuous Itô diffusion processes, as happens for other heteroscedastic formulations. One of the schemes developed in the paper-the Volatility-switching ARCH-differs from the existing asymmetric models insofar as it is able to capture a particular aspect of the behaviour of the volatilities, i.e. the reversion of their asymmetric reaction to news. Empirical evidence from stock market returns in six countries shows that such a model outperforms traditional asymmetric ARCH equations.