Stroke: The combination of acrocyanosis (which can be a sign of decreased peripheral circulation, possibly due to immobilization or autonomic dysfunction) and left hemiplegia strongly suggests a stroke, particularly one affecting the right hemisphere of the brain, which could lead to left-sided weakness or paralysis.

Multiple Sclerosis: Although less common, multiple sclerosis (MS) could potentially cause both hemiplegia (through spinal cord or brainstem lesions) and acrocyanosis (if there's significant autonomic nervous system involvement affecting vascular tone).
Spinal Cord Injury: A high spinal cord injury could result in both hemiplegia (depending on the level and completeness of the injury) and acrocyanosis due to autonomic dysfunction affecting peripheral blood flow.

Subclavian Steal Syndrome: This condition, characterized by reversal of blood flow in the vertebral artery due to significant stenosis or occlusion of the subclavian artery proximal to the origin of the vertebral artery, can lead to neurological symptoms (including hemiplegia if the brain is affected) and decreased blood flow to the arm, potentially causing acrocyanosis.
Aortic Dissection: Although more commonly associated with chest or back pain, an aortic dissection could lead to decreased blood flow to various parts of the body, including the brain (causing hemiplegia) and limbs (potentially causing acrocyanosis), and is a medical emergency.

Buerger's Disease (Thromboangiitis Obliterans): A rare disease of the blood vessels in the arms and legs, which could potentially cause acrocyanosis due to decreased blood flow. If there's central nervous system involvement (rare), it could also lead to hemiplegia.
Moyamoya Disease: A rare condition caused by narrowing or blockage of the internal carotid artery and its branches, which could lead to stroke-like symptoms, including hemiplegia. Acrocyanosis might be less directly related but could occur in the context of widespread vascular disease.