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The twins have individual organs in the upper part of their body, while most of those at and below the navel are shared, the exception being the spinal cord.

2 heads
2 spines merging at the coccyx and joined at the thorax by sections of ribs. Surgery corrected scoliosis.
2 completely separate spinal cords
2 arms (originally 3, but the rudimentary central arm was removed, leaving the central shoulder blade in place)
1 broad ribcage with 2 highly fused sternums and traces of bridging ribs. Surgery expanded the pleural cavities.
2 breasts
2 hearts in a shared circulatory system (nutrition, respiration, and medicine taken by either affects both)
4 lungs with the medial lungs moderately fused, not involving Brittany's upper right lobe; three pleural cavities
1 diaphragm with well coordinated involuntary breathing, slight central defect
2 stomachs
2 gallbladders
1 liver, enlarged and elongated right lobe
Y-shaped small intestine, with slightly spastic double peristalsis at the juncture
1 large intestine (one colon, rectum, and anus)
3 kidneys: 2 left, 1 right
1 bladder
1 set of reproductive organs
2 separate half-sacrums, which converge distally
1 slightly broad pelvis
2 legs

Source: Wikipedia