  We construct an order reconstruction (OR)-type Landau-de Gennes critical
point on a square domain of edge length $\lambda$, motivated by the well order
reconstruction solution numerically reported by Kralj and Majumdar. The OR
critical point is distinguished by an uniaxial cross with negative scalar order
parameter along the square diagonals. The OR critical point is defined in terms
of a saddle-type critical point of an associated scalar variational problem.
The OR-type critical point is globally stable for small $\lambda$ and undergoes
a supercritical pitchfork bifurcation in the associated scalar variational
setting. We consider generalizations of the OR-type critical point to a regular
hexagon, accompanied by numerical estimates of stability criteria of such
critical points on both a square and a hexagon in terms of material-dependent
constants.
