Case study and speculations on the brain of Edward Mordake
Mordake was a conjoined twin fused at the head (Fig. 1). Twins conjoined at the heads are rare, occurring only once in every 2.5 million live births. Moreover, in Mordake's case it was a parasitic twin head with an undeveloped body that was fused with his own head. This condition is referred to as craniopagus parasiticus, or épicome / epicomus. Craniopagus parasiticus is an extremely rare teratology, of which only six cases have been recorded in the medical literature (Bondeson et al, 1989). 


Fig. 1. Wax reproduction from Edward Mordakes face. Source: Bosmia et al., (2014). 

The story of Edward Mordake is, however, obscure and met with skepticism. Moreover, no medical details are known as far as I know. However, he was reported to be intelligent and healthy, just depressed by the second parasitic head (Bosmia et al, 2014). To me it appears a story of a person with a normal brain and a parasitic face attached to his head. However, there are rumors that the parasitic head could talk independently, leaving open the possibility that the parasitic head may have contained a separate brain, or shared Mordake's own brain. 

More is known about the double-headed Bengal of Everard Home (Fig. 2).

 
Fig. 2. Skull of the double-headed Bengal (museum exhibit, left) and an artist's impression of the case (right). Source: Bondeson et al, (1989)  

The natural head and body were perfectly normally developed, but a number of anomalies were noted on an examination of the parasitic head, such as excessive tearing and the absence of pupil reflexes. Upon dissection the parasitic head was shown to show various anomalies, such as missing anatomical elements (e.g. auditory canal) or underdeveloped structures (e.g., small lower jaw bones).  Importantly, the brains were separate from each other and covered in individual and proper coverings. The dura mater of each brain adhered firmly and contained
many large vessels. Bondeson et al. (1989) describe the remainder of the six cases as well, which were still-borns or lived just very shortly. They had completely separated brains too. 

References
- Bondeson et al, Surg Neurol (1989); 31:426-34
- Bosmia et al, Childs Nerv Syst (2014)