This paper uses a Dependence-Switching Copula (DSC) to examine the dependence and tail dependence between oil prices and stock returns of 54 clean energy companies in the WilderHill Clean Energy Index (WECI) and 100 technology companies in the NYSE Arca Technology Index (NATI) under four different market conditions, namely, bullish WECI or NATI/high oil price, bearish WECI or NATI/low oil price (i.e., positive correlation regimes), bullish WECI or NATI/low oil price, and bearish WECI or NATI/high oil price (i.e., negative correlation regimes). The findings reveal a asymmetric dependence structure under the positive correlation regimes, while a symmetric dependence under negative correlation regimes. However, the results also show a dissimilarity in the responses of stock returns of the selected companies with respect to the fluctuation in the oil prices. The tail dependence between oil and clean energy indices is the lowest under the positive correlation regimes and the that between oil and technology indices is the lowest under negative correlation regimes. For the two considered pairs, we note a dominance of the chasing effect (negative dependence) for some periods, while a dominance of the portfolio rebalancing effect (positive dependence) for the other periods. Our findings provide crucial information to investors and portfolio managers seeking low-risk opportunities in distressing times.