  Spontaneous orbital magnetism observed in twisted bilayer graphene (tBG) on
nearly aligned hexagonal boron nitride (BN) substrate builds on top of the
electronic structure resulting from combined G/G and G/BN double moire
interfaces. Here we show that tBG/BN commensurate double moire patterns can be
classified into two types, each favoring the narrowing of either the conduction
or valence bands on average, and obtain the evolution of the bands as a
function of the interlayer sliding vectors and electric fields. Finite valley
Chern numbers $\pm 1$ are found in a wide range of parameter space when the
moire bands are isolated through gaps, while the local density of states
associated to the flat bands are weakly affected by the BN substrate invariably
concentrating around the AA-stacked regions of tBG. We illustrate the impact of
the BN substrate for a particularly pronounced electron-hole asymmetric band
structure by calculating the optical conductivities of twisted bilayer graphene
near the magic angle as a function of carrier density. The band structures
corresponding to other $N$-multiple commensurate moire period ratios indicate
it is possible to achieve narrow width $W \lesssim 30$ meV isolated folded band
bundles for tBG angles $\theta \lesssim 1^{\circ}$.
