Morphologically, Catdog’s body is highly asymmetrical. The primary heart is located on the feline side of the organism, with a small vestigial equivalent on the side of the canine head. Alternatively, the lungs and diaphragm are shifted to the side of the canine. Catdog contains a single functional version of every necessary organ, often with an equivalent but inactive version on the opposite side of the body. This accounts for the creature’s ability to function with only a single body cavity shared between two heads. Although one head is characterized as feline, and the other is distinctly canine, the body between the two is indistinct and difficult to categorize. It does not closely match either head’s ordinary animal anatomy, and instead contains an extremely flexible spine and four blunt paws of identical size and shape. Each head controls the set of limbs closest to it, but does not share sensory data to coordinate movement. Locomotion is therefore laborious and often directionless, often with whichever head is moving forward dragging the entire body behind it. 

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EDIT

Wow, this piece is really taking off! And a lot of commenters and direct messages have been excitedly asking-
WHY DOESN'T CATDOG JUST HAVE A 100% EFFICIENT DIGESTIVE SYSTEM AND NEVER NEED TO POOP??

This is an interesting (but impossible) idea. And I looooooooooove absurd scientific questions and hypothetical biology, so I thought I'd take some time to explain why this explanation doesn't work. 

Long story short- Digestion isn't magic, and matter cannot be created or destroyed. 

In vertebrate digestion, large chemical structures are broken down into small water-soluble food molecules so that they can be absorbed by the body. Generally, Fat, protein, and carbohydrates are the primary sources of nutrients for most higher life forms. 

- Proteins are broken down into polypeptides that are then broken down into amino acids. 
- The short chain lipids in fat are broken down into diglycerides, and the digestion of one molecule of fat (a triglyceride) results a mixture of fatty acids, mono- and diglycerides.
- Carbohydrates like dietary starches are composed of glucose, which are units arranged in long polysaccharide chains called amylose. During digestion, bonds between glucose molecules are broken and turned into progressively smaller chains of glucose. This results in simple sugars that can be absorbed by the small intestine.

That's a lot of scientific jargon, but essentially what it means is that digestive processes break down chemical structures, but both the useable and unusable components of those structures remain present in the body. Not to mention totally indigestible things like pectins, cellulose, and other dietary fibers which are essentially inert matter with no chemical energy to utilize. These components need to go somewhere. 

That's not even really getting started on the waste products that the body's own metabolic processes produce independently of digestion. Which I won't go to far into. Suffice it to say- cellular metabolism generates many by-products which are rich in nitrogen and essentially toxic. Urea, uric acid, creatinine, and many other byproducts must be cleared from the bloodstream routinely (which is mostly why we pee and poop).

So... maybe Catdog has a totally alien internal anatomy that violates all biological principals as we understand them?  
You might say. 

Well, since we see Catdog eating human/animal food at a rate pretty consistent with an ordinary animal its size this seems pretty silly.... Where does that mass go? Why does it eat sandwiches if it doesn't perform normal digestion? 

Even if Catdog had some kind of immensely powerful reactor in its stomach that could perform successful nuclear fission this wouldn't work. Fission isn't really turning matter into energy. It just releases the binding energy of the nucleus. This binding energy is part of the measured mass pf the nucleus, and if you want to separate "matter" and "energy" (not really possible), then it counts as energy. However, the number of protons and neutrons that you start with in fission doesn't change at the end. From this POV, where particles count as "mass", we can say that no mass was created or destroyed, and only the nuclear binding energy was released. These used particles still need to go somewhere.

I like the idea that Catdog basically violates the laws of physics by never pooping, because I'm interested in hypothetical scientific systems, and enjoy working backwards from absurd fictional premises to potential real world conclusions. 
But alas... as the classic children's book clearly states....

Everybody poops. 

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