Twisted graphene multilayers exhibit strong electronic correlations leading to a range of experimental signatures. Yet how these signatures relate to each other and to the microscopic ground states—and how twist angle and band structure reshape them—remains poorly understood. Here we study this interplay by correlating local thermodynamic and transport measurements in twisted trilayer graphene with unequal angles and flat electronic bands. We use a scanning single-electron transistor to map the impact of electron–electron interactions in the sample by selecting a region with smooth local twist angle evolution. We observe gapped correlated insulator states and asymmetric oscillations in the inverse electronic compressibility, both exhibiting pronounced electron–hole asymmetry with distinct ‘magic’ angles for conduction and valence bands. Subsequent transport measurements in the same region reveal robust superconductivity with a similar electron–hole asymmetry. Our measurements indicate that superconductivity is not directly tied to the correlated insulator states. Instead, its critical temperature correlates closely with the strength of the compressibility oscillations, suggesting a common origin or link between the two. By combining a local probe with transport measurements, we uncover connections between superconductivity and thermodynamic correlation signatures that are not apparent from either technique in isolation. These findings highlight the power of our dual approach.