In this paper, we examine how to quantify asymmetries in volatility spillovers that emerge due to bad and good volatility. Using data covering most liquid U.S. stocks in seven sectors, we provide ample evidence of the asymmetric connectedness of stocks at the disaggregate level. Moreover, the spillovers of bad and good volatility are transmitted at different magnitudes that sizably change over time in different sectors. While negative spillovers are often of substantial magnitudes, they do not strictly dominate positive spillovers. We find that the overall intra-market connectedness of U.S. stocks increased substantially during the recent financial crisis. •We suggest how to quantify asymmetries in volatility spillovers.•Asymmetries emerge due to bad and good volatility.•Asymmetric connectedness is evidenced for most liquid U.S. stocks in seven sectors.•Spillovers are transmitted at magnitudes that change over time in different sectors.•Negative spillovers do not strictly dominate positive spillovers.