There are few dual neurology/psychiatry residencies so if you really want to practice both you can. Some people also do psych training after neuro (doesn't make sense the otherway around as will be too long!) Otherwise you can do psych and then do a neuropsychiatry fellowship. Incidentally, movement disorders fellowships are often open to psychiatrists. If you do neuro, you can also do a cognitive and behavioral neurology fellowship.
I am a neuropsychiatrist. I specialize in brain-behavior relationships in the context of function and dysfunction in the brain. We can perform physical examinations, neurobehavioral examinations, make medical diagnoses, order and interpret brain scans, laboratory tests, and neurophysiological tests, prescribe (and de-prescribe) medications, and perform psychotherapy. Common neuropsychiatric disorders including the dementias (Alzheimer's, bvFTD, DLB, vascular dementia etc), TBI, persistent post-concussive symptoms, functional neurological disorders, chronic pain, amnestic disorders, and psychiatric manifestations of neurological disease (e.g. epilepsy, movement disorders, autoimmune encephalitis, stroke, neurogenetic disorder, MS).
I'll give you and idea of the sorts of cases I might see or get consulted on for an expert opinion:
An patient with parkinsonism, paranoia, and hallucinations
An older woman patient with a history of breast cancer and no prior psychiatric history presenting with catatonic symptoms and episodic memory loss
A young women with episodes of unexplained amnesia where she does not recognize her children or her own home
A young woman with a history of extensive childhood sexual trauma with recurrent orthostasis and fainting spells
An autistic patient with persistent headaches, dizziness, photophobia, phonophobia, sleeplessness, panic attacks following a head injury w/o LOC from an MVA
A patient who presented with sudden onset unexplained stroke-like symptoms who now speaks in a slow British Accent
A patient with MELAS with unexplained episodic hemiparesis
A patient with stiff person syndrome with severe anxiety, exaggerated startle response, and abnormal gait
A patient with severe intellectual disability and autism with self-injurious behaviors
A patient who was diagnosed with normal pressure hydrocephalus s/p VP shunt placement now experiencing cognitive decline, paranoia, and apathy
An older patient with parkinson's disease who sees the devil and tries to kill their husband
A patient with refractory epilepsy who has multiple "rage attacks" with amnesia without ictal correlate for which the police are frequency called
An older patient with Alzheimer's who believes that their husband and father is the same person
You do get some zebras and it can be very intellectually stimulating. But it can be hard work regardless of whether you are on the more psychological side (vicarious traumatization and dealing with personality disorders) or neurodegenerative side (dealing with angry relatives who won't accept why you can't or won't 'fix' their demented loved one). Doing this work properly requires time and medicine rewards volume. Which is another way of saying that most healthcare systems do not value this work. There are very few advertised jobs for neuropsychiatrists. And for historical reasons, about half of the behavioral neurologists and neuropsychiatrists in the US are in Boston.