Lattice reconstruction in twisted transition metal dichalcogenides manifest in intrinsic asymmetry of electronic wavefunctions for 3R homo-bilayers and strong piezoelectric textures in 2H homo-bilayers. Van der Waals heterostructures form a unique class of layered artificial solids in which physical properties can be manipulated through controlled composition, order and relative rotation of adjacent atomic planes. Here we use atomic-resolution transmission electron microscopy to reveal the lattice reconstruction in twisted bilayers of the transition metal dichalcogenides, MoS2 and WS2. For twisted 3R bilayers, a tessellated pattern of mirror-reflected triangular 3R domains emerges, separated by a network of partial dislocations for twist angles theta < 2 degrees. The electronic properties of these 3R domains, featuring layer-polarized conduction-band states caused by lack of both inversion and mirror symmetry, appear to be qualitatively different from those of 2H transition metal dichalcogenides. For twisted 2H bilayers, stable 2H domains dominate, with nuclei of a second metastable phase. This appears as a kagome-like pattern at theta approximate to 2 degrees, transitioning at theta -> 0 to a hexagonal array of screw dislocations separating large-area 2H domains. Tunnelling measurements show that such reconstruction creates strong piezoelectric textures, opening a new avenue for engineering of 2D material properties.