Next time you're brushing your teeth, take a look in the mirror. You will see a person like you, except that they appear to be using their opposite hand. Now notice that your body's top and bottom are the same as theirs...curious, isn't it? Rotate your head 90 degrees to one side, and these relations are preserved. Why?
The mirror creates a 'mirror world' in which the direction toward and away from the mirror is reversed. The mirror image of a bilaterally symmetric human body, if interpreted as existing in non-mirror or 'real' space, appears to have left and right switched because they look similar enough to be interchanged. We don't confuse top/bottom and front/back distinctions because they are unambiguous.
Interestingly, left/right distinctions are made after front/back and top/bottom ones in vertebrate embryogenesis, and are sometimes mistaken. About one in 60,000 humans develop with a condition called situs inversus, in which the placement of their heart and all the other asymmetric internal organs are on the 'wrong' side. Such choices of asymmetries are called symmetry breaking.
The disruption of certain genes causes vertebrate embryos to display 'left' or 'right' body asymmetries with 50/50 probability, removing the usual bias in symmetry breaking. The mechanism of asymmetry depends on cilia spinning on the embryo's surface, directing extra-embryonic fluid in a particular left-right direction, thus affecting further development. This, in turn, depends on the chirality of molecules used to build cilia and their motors that spin in one direction and not the other.
This example of biological asymmetry depends on a choice of chemical asymmetry for its bookkeeping. On a much broader scale, physical laws also distinguish right from left, breaking cosmic symmetries to produce left-handed and right-handed force relations, and more matter than antimatter.
The latter effect could be important if you're a vertebrate.
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