Astrophysical black holes (BHs) have two fundamental properties: mass and spin. While the mass-evolution of BHs has been extensively studied, much less work has been done on predicting the distribution of BH spins. In this paper we present the spin evolution for a sample of intermediate-mass and massive BHs from the newHorizon simulation, which evolved BH spin across cosmic time in a full cosmological context through gas accretion, BH-BH mergers and BH feedback including jet spindown. As BHs grow, their spin evolution alternates between being dominated by gas accretion and BH mergers. Massive BHs are generally highly spinning.Accounting for the spin energy extracted through the Blandford-Znajek mechanism increases the scatter in BH spins, especially in the mass range $10^{5-7} \rm \ M_\odot$, where BHs had previously been predicted to be almost universally maximally spinning. We find no evidence for spin-down through efficient chaotic accretion. As a result of their high spin values, massive BHs have an average radiative efficiency of $<\varepsilon_{\rm r}^{\rm thin}> \approx 0.19$. As BHs spend much of their time at low redshift with a radiatively inefficient thick disc, BHs in our sample remain hard to observe. Different observational methods probe different sub-populations of BHs, significantly influencing the observed distribution of spins. Generally, X-ray-based methods and higher luminosity cuts increase the average observed BH spin. When taking BH spin evolution into account, BHs inject on average between 3 times (in quasar mode) and 8 times (in radio mode) as much feedback energy into their host galaxy as previously assumed.