There are a number of possibilities, from benign to serious.
Sweating is under the control of the sympathetic nervous system, as is vasodilation (facial redness). Anhydrosis of only one side of the face would be due dysfunction of the trigeminal nerve (which also carries sensory and motor components) on that side of the face. The trigeminal nerve can lose function for many reasons, but rarely does it lose only one of it's functions (i.e. the physical exam should help here.)
Ross syndrome is a rare disorder of unknown etiology characterized by segmental an-/hypohydrosis, pupil irregularity (called Adie's pupil), and diminished deep tendon reflexes.
Harlequin syndrome is another possibility.
Horner's syndrome is the broad term for the decreased sympathetic nerve function, and, again, involves more than anhydrosis. It is characterized by the triad of miosis (constricted pupil), partial ptosis, and loss of hemifacial sweating. Horner's is not rare per se, and can occur alone or with a number of illnesses.
Horner syndrome can be congenital, acquired, or purely hereditary (autosomal dominant). The interruption of the sympathetic fibers may occur centrally (ie, between the hypothalamus and the fibers’ point of exit from the spinal cord [C8 to T2]) or peripherally (ie, in cervical sympathetic chain, at the superior cervical ganglion, or along the carotid artery).
Serious disorders should be ruled out, e.g. internal carotid artery dissection (not always symptomatic), malignancies anywhere from central nervous system/trigeminal/apical lung/etc., other causes of carotid ischemia, etc. Since the differential of Horner's Syndrome is large and it is a neurological problem, a neurologist is probably the best bet for a diagnosis here.
Although Horner syndrome is commonly an incidental finding related to a benign cause, it occasionally may be a manifestation of a serious and life-threatening disorder. Careful direction of the history to rule out such life-threatening disorders is vital.
Adie's Pupil in the Ross Syndrome (Good picture here.)
Ross Syndrome with Sweating Anomaly Associated with Sjögren Syndrome: An Infrared Thermographic Case Study
Harlequin syndrome - one face of many etiologies (Also a good picture, and open access.)