What is a rainbow without color called?

A fogbow is a faint arc of white in a bank of fog, when the sun is (low in the sky) behind you and the fog is in front of you.

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Moreover, can there be an upside down rainbow?

They're called circumzenithal arcs, and they're not really rainbows. Instead, they're caused by ice crystals in the upper atmosphere. These arcs are related to the frequently seen halos around the sun or moon. It is often described as an 'upside-down rainbow' by first-timers. What is the rarest rainbow? Twinned rainbow. The rarest type of rainbow start from the same base but them split along the arc to form a primary and secondary rainbow. Twinned rainbows are formed when sunlight is refracted after coming into contact with two rain showers which have different size of droplets from each other.

Keeping this in consideration, why is the second rainbow inverted?

It occurs when refracted light does not escape the raindrop after being reflected the first time. Instead, the refracted light reflects off the raindrop's surface a second time as well, producing a secondary rainbow with its colors reversed compared to the primary rainbow. You can also ask how rare is it to see a double rainbow? So the answer to your question is 'no', it isn't rare at all, in fact a secondary arc is always present somewhere, but is not always visible to the observer for reasons of geometry and background light intensity.