Food fat content is one of the most controversial factors from a consumer’s point of view. Aim: (1) The trends in consumer attitudes towards pork and the fat and meat compositions in Duroc and Altai meat breeds and Livny and Mangalitsa meat and fat breeds were studied. (2) Methods: Netnographic studies were used to assess Russian consumer purchasing behavior. Protein, moisture, fat, backfat fatty acid content from pigs, longissimus muscles, and backfat from (A) Altai, (L) Livny, and (M) Russian Mangalitsa breeds were compared with those from (D) Russian Duroc. Raman spectroscopy and histology were applied to the backfat analysis. (3) Results: The attitude of Russian consumers to fatty pork is contradictory: consumers note its high fat content as a negative factor, but the presence of fat and intramuscular fat is welcomed because consumers positively associate them with better taste, tenderness, flavor, and juiciness. The fat of the ‘lean’ D pigs did not show a “healthy” fatty acid ratio, while the n-3 PUFA/n-6 PUFA ratio in the fat of the M pigs was the best, with significant amounts of short-chain fatty acids. The highest UFA content, particularly omega 3 and omega 6 PUFA, was found in the backfat of A pigs with a minimum SFA content. The backfat of L pigs was characterized by a larger size of the adipocytes; the highest monounsaturated and medium chain fatty acid contents and the lowest short-chain fatty acid content; the ratio of omega 3 to omega 6 was 0.07, and the atherogenicity index in L backfat was close to that of D, despite the fact that D pigs are a meat type, while L pigs are a meat and fat type. On the contrary, the thrombogenicity index in L backfat was even lower than the D one. (4) Conclusions: Pork from local breeds can be recommended for functional food production. The requirement to change the promotion strategy for local pork consumption from the position of dietary diversity and health is stated.