They have red flesh, with white flesh just under the skin.
These are floury potatoes.
Possibly the same potato as “Red Salad”, “Beetroot Tattie”, and “Egyptian Red.” 
The flesh retains the colour when cooked. Despite the colour, they make fluffy mash.
Highland Burgundy Potatoes have possibly been known since 1936:
“I was approached at the East Anglian Potato Event by Gordon and Sue Baldam, then of Nordelph, Norfolk, who gave me my first, very sickly, tuber. They had been given the variety by friends who had received it from Mr Harris, an elderly HDRA member. He, in turn, had obtained it in the early 1970s from the eighty-year-old grandsom of a former Royal Botanic Gardens, Kew curator. There is supposed to be a newspaper clipping in existence which tells of the variety being used in 1936 for a special dinner at the Savoy in honour of the Duke of Burgundy. This is convoluted but believable partial provenance for the name anyway.” 
 Romans, Alan. The Potato Book. London: Frances Lincoln Ltd. 2005. Page 120.