Let me make one thing clear GARCH is not an Assymetric type model so for modeling assymetric behavior go for ARMA-APARCH-MT OR T-GARCH type model.

If you just want to verify if volatility reacts in a different way to shocks depending on their sign you can see that directly from the estimation of an EGARCH (provided you have enough data to estimate the model)

Alternatively, you estimate a simple model, like GARCH, and test with a sign bias test if you should have included an asymmetric term in the conditional variance specification, that is as in EGARCH, APARCH, TARCH, and GJR, among others.