Long memory, asymmetry, volatility spillover aspects of the realized volatilities (RVs) of the log returns of the Korean KOSPI, the Korean won/US dollar exchange rate, Korean treasury bond futures, and the US S&P 500 index are investigated in this paper. For all RVs, significant long memories and asymmetries are identified, which can improve forecasts if suitably used. A new long memory asymmetric threshold HAR-RV (heteroskedastic autoregressive realized volatility) model is proposed. The new model shows explicit differences between the coefficients of daily, weekly, and monthly RVs for days of positive returns and those for days of negative returns. By analyzing the RVs, we find that pseudo out-of-sample forecast performance of the proposed model is better than a widely used existing asymmetric long memory model in about half of cases considered. The new model as well as another existing asymmetric model show that forecast improvements due to implied volatilities for the S&P 500 index decrease when asymmetries in RVs are accounted for. The two asymmetric models find significant decompositions of the spillover effects from the influential market S&P 500 index to the influenced market KOSPI (or from the KOSPI to the Korean won/US dollar exchange rate) into three oscillating components: substantial positive short-term daily spillovers, moderate negative weekly spillovers, and somewhat positive long-term monthly spillovers.