Surface reconstructions can be divided into two groups:
displacement and misplacement reconstructions.

Compared to the unreconstructed configuration, particles on the surface are moved only locally or are transported away completely.

Misplacement reconstructed surfaces have typically a different average surface height than their unreconstructed counter parts. The top layer(s) can be interpreted as only partially completed and have long range positional order.

displacement reconstruction with topological defects

Above the deconstruction phase transition temperature, misplacement reconstructed surfaces become disordered flat (DOF). The top layer(s) are still only partially completed but now lack positional order.

The DOF surface structure is not considered rough. Surface roughness is a large scale phenomena where the surface interface width measured over a length scale L diverges with L. From a surface roughening perspective, a DOF phase is a step liquid with long range topological up-down order.

A preroughening transition is a phase transition from an ordered flat phase into a DOF phase.

Surface roughening and reconstruction transitions can compete. Roughness can be compatible or incompatible with the reconstruction depending on topological details of the crystal structure and the type of reconstruction.

For example, the roughening and reconstruction degrees of freedom decouple in missing row reconstructed simple cubic (110) facets; such that a so-called reconstructed rough phase is possible.

The same degrees of freedom couple strongly in FCC (110) facets like Au(110) and Pt(110) , such that surface roughening can not take place before the reconstruction order parameter vanishes. Roughening induces a simultaneous deconstruction transition if the surface does not deconstruct before it roughens.

Another example is the Si(001) type 1x2 reconstruction; (pre-)roughening must induce a simultaneous deconstruction transition.