This research reports on the effects of including natural pozzolans in two Portland cements with different mineralogical compositions, with and without excess gypsum at amounts equivalent to 7.0% SO3. The main analytical techniques used to study these effects were: the amount of water needed to make a paste of normal consistency, the 2-day Frattini pozzolanicity test and conduction calorimetry. The results obtained showed that these natural pozzolans caused contradictory (accelerating and retarding) effects on the rheology of the resulting cements, depending on the mineralogical composition of the respective Portland clinkers as well as the reactive chemical composition of the pozzolans, in particular their reactive alumina content (Al2O3 r−). The addition of gypsum also caused acceleration and delays in the calorimetric evolution of the resulting pastes, which proved to be heavily dependent upon the more or less aluminic chemical character of the natural pozzolans studied. This, in turn, was conditioned by the higher or lower Al2O3 r− content (for the SiO2 r− content was of a very similar order of magnitude in all three pozzolans analyzed). The Al2O3 r− content was likewise responsible for paste behaviour in the afore-mentioned trials and analyses, and the pozzolanic activity exhibited by the compound was found to be more specific than generic, indirectly stimulating C3A hydration more intensely and rapidly than C3S hydration in PC1, one of the two Portland cements used. Indeed, when these natural pozzolans exhibited such prior pozzolanic activity in the second cement studied, PC2, the hydration of its 79.5% of C3S was not indirectly stimulated to the same degree; rather, the contrary effect was observed, i.e., this cement was physically diluted by the three pozzolans. Pozzolan O stimulated hydration directly and non-directly more than indirectly, while pozzolan C acted conversely, and A exhibited varying combinations of the two patterns. The physical state of the reactive alumina, Al2O3 r−, in these three natural pozzolans, must be more amorphous than vitreous, i.e., resembling metakaolin more than fly ash in this regard. That notwithstanding, the reactive alumina content in each pozzolan must have conditioned the water/cementitious material ratio obtained for the respective blends with both types of Portland cement (a finding that could be used in future for speedy, simple, reliable and economical characterization), as well as their specific pozzolanicity developed and the rate and total heat of hydration generated by such blended cements.