Why revolutions (or turns) are dimensionless? I think that the reason is because one revolution or one turn is equal to $2 \pi$ rad or to $360$ degrees.
We can relate rads and degrees to two units of length that cancel each other.
rad $= \frac{arc\: length}{radius\: of\: the \:arc\: length}$ 
degree $=$ arc length$ * \frac {1}{360}$ of the total circunference.
In both cases the meters from the numerator cancel with the meters from the denominator. This implies that rads and degrees are dimensionless, but not unitless.
Is there another explanation why a revolution is dimensionless?
Is there an analogous explanation, that meters with meters cancel each other, for revolutions?
Or you can only explain it equating revolutions with degrees or radians?
Morover, Tipler's Physics for scientists and engineers explains what a dimension is in this way.

I can measure the number of revolutions (for instance with a photoelectric sensor) of a turning plate. So I have a number with units(revolutions or rads). Don't we have a dimension in this case, the angle? 
 A: 
Is there another explanation why a revolution is dimensionless?

In the end the radian is dimensionless because the BIPM (the organization which defines the SI) decided that it is dimensionless. 
Here is the official definition of the SI, updated earlier this year: https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf 
On p 136 it defines each of the base units to have its own unique dimension, and specifies that all derived units have dimensions corresponding to the base units used to derive the derived unit. Then on p 137 it defines the radian to be a derived unit of m/m, implying that the radian is dimensionless.
The dimensionality of a unit is just as much a matter of convention as its size. For instance, in SI the ampere is a fundamental unit with dimension of current, $I$ meaning that charge has dimensions of $IT$. In contrast, in cgs units the statcoulomb has dimensions of $L^{3/2}M^{1/2}T^{-1}$. 
So, although the radian is defined by the BIPM as dimensionless, there would be nothing logically wrong with a non-SI unit of angle that was considered to have dimensions. It is entirely a matter of convention. However, note that if you change your units then you may also need to change some of your physics formulas. 
You have specifically asked about “turns” or “revolutions” rather than radians. As far as I am aware there is no governing body defining a system of units in which the unit of angle is a turn or a revolution. Therefore, the dimensionality is entirely up to you. If you like you may consider a turn to have dimension, and if you like you may consider it to be dimensionless. There is nothing which physically or mathematically prohibits either convention. 
