Why is a cylinder not perfectly symmetric as a sphere?

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I have read that a cylinder not being perfectly symmetric is the reason behind Rayleigh instability: the process that makes bubbles out of a stream of water.

But a cylinder seems also perfectly symmetric to me

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Suppose to have a sphere of crystal. You can rotate the sphere in every direction and it will always look the same and you will never be able to tell if it's up or down (the question doesn't even make sense). Instead take a crystal glass with the shape of a cylinder. You will always be able to tell if the glass is standing up or laying down on the table.

This is because you can switch the axis $x,y,z$ and the sphere will appear always the same, while with the cylinder you will always have an axis which is different from the other two.