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Spontaneous orbital magnetism observed in twisted bilayer graphene (tBG) onnearly aligned hexagonal boron nitride (BN) substrate builds on top of theelectronic structure resulting from combined G/G and G/BN double moireinterfaces. Here we show that tBG/BN commensurate double moire patterns can beclassified into two types, each favoring the narrowing of either the conductionor valence bands on average, and obtain the evolution of the bands as afunction of the interlayer sliding vectors and electric fields. Finite valleyChern numbers $\pm 1$ are found in a wide range of parameter space when themoire bands are isolated through gaps, while the local density of statesassociated to the flat bands are weakly affected by the BN substrate invariablyconcentrating around the AA-stacked regions of tBG. We illustrate the impact ofthe BN substrate for a particularly pronounced electron-hole asymmetric bandstructure by calculating the optical conductivities of twisted bilayer graphenenear the magic angle as a function of carrier density. The band structurescorresponding to other $N$-multiple commensurate moire period ratios indicateit is possible to achieve narrow width $W \lesssim 30$ meV isolated folded bandbundles for tBG angles $\theta \lesssim 1^{\circ}$.

Electron-hole asymmetry and band gaps of commensurate double moire patterns in twisted bilayer graphene on hexagonal boron nitride