We present sub-kpc-scale mapping of the 870 $\mu$m ALMA continuum emission in six luminous ($L_{\rm IR}~\sim~5~\times10^{12}$ L$_{\odot}$) submillimeter galaxies (SMGs) from the ALESS survey of the Extended Chandra Deep Field South. Our high-fidelity 0.07$''$-resolution imaging ($\sim$500 pc) reveals robust evidence for structures with deconvolved sizes of $\lesssim$0.5-1 kpc embedded within (dominant) exponential dust disks. The large-scale morphologies of the structures within some of the galaxies show clear curvature and/or clump-like structures bracketing elongated nuclear emission, suggestive of bars, star-forming rings, and spiral arms. In this interpretation, the ratio of the `ring' and `bar' radii (1.9$\pm$0.3) agrees with that measured for such features in local galaxies. These potential spiral/ring/bar structures would be consistent with the idea of tidal disturbances, with their detailed properties implying flat inner rotation curves and Toomre-unstable disks (Q