Metamaterials constructed from origami units of different types and behaviors could potentially offer a broader scope of mechanical properties than those formed from identical unit types. However, the geometric design rules and property programming methods for such metamaterials have yet to be extensively explored. In this paper, we propose a new kind of origami metasheet by incorporating a family of different square-twist units. The tessellation rule of these metasheets is established to allow compatible mountain–valley crease assignments and geometric parameters among neighboring units. We demonstrate through experiments that the energy, initial peak force, and maximum stiffness of the metasheets can be obtained by a summation of the properties of the constitutional units. Based on this, we are able to program the mechanical properties of the metasheets over a wide range by varying the types and proportions of the units, as well as their geometric and material parameters. Furthermore, for a metasheet with a fixed number of units, all the geometrically compatible tessellations can be folded out of the same pre-creased sheet material by simply changing the mountain–valley assignments, thereby allowing the properties of the metasheet to be re-programmed based on specific requirements. This work could inspire a new class of programmable origami metamaterials for current and future mechanical and other engineering applications.