Nanodroplet-mediated histotripsy (NMH) is an ultrasound ablation technique combining histotripsy with acoustically sensitive perfluorocarbon (PFC) nanodroplets that can be selectively delivered to tumor cells for targeted tumor ablation. NMH takes advantage of the significantly reduced cavitation threshold of the nanodroplets, allowing for cavitation to be selectively generated only in regions containing nanodroplets. Understanding the physical mechanisms underlying the nanodroplet cavitation process is essential to the development of NMH. In this study, we hypothesize that cavitation nucleation is caused by the negative pressure (p-) exposed to the PFC, and the NMH cavitation threshold is therefore determined by the incident p- of the single-cycle pulses commonly used in NMH. This paper reports the first study that separately investigates the effects of negative and positive pressure on the NMH cavitation threshold using near half-cycle ultrasound pulses with dominant negative (negative-polarity pulses) or positive (positive-polarity pulses) pressure phases. Tissue phantoms containing perfluorohexane (PFH) nanodroplets were exposed to negative-polarity and positive-polarity pulses generated by a frequency compounding transducer recently developed in our lab, and the probability of generating cavitation was measured as a function of peak negative (p-) and peak positive (p+) pressure. The results showed close agreement in the p-cavitation threshold for PFH phantoms exposed to negative-polarity (11.4 +/- 0.1 MPa) and positive-polarity (11.7 +/- 0.2 MPa) pulses. The p+ at the cavitation threshold, in contrast, was measured to be significantly different for the negative-polarity (4.0 +/- 0.1 MPa) and positive-polarity (42.6 +/- 0.2 MPa) pulses. In the final part of this study, the experimental results were compared to the cavitation threshold predicted by classical nucleation theory (CNT), with results showing close agreement between simulations and experiments. Overall, the results support our hypothesis and provide significant insight into the physical mechanisms underlying NMH.