Can you compare Cramer's V and Spearman's $\rho$ values? I did correlation tests between a number of variables consisting of both interval and categorical scales ($n = 1274$, one sample). I used $\chi ^2$ test for categorical vs ordinal and Spearman's $\rho$ for interval vs ordinal data. 
The correlation between climate zone (zone $1, 2, 3, 4$) and daily exposure to air-conditioning (low, medium, high) is $0.35$ (Cramer's V). 
The correlation between exposure to AC and pro-environmental attitude score ($1-5$, interval) is $0.21$ (Spearman's $\rho$). 
Can I compare the strength of the relationship if I used different correlation tests, say the relationship between climate and exposure to AC is stronger than that between the pro-environmental attiudes and exposure to AC?
 A: Up to a point.  The closer you get to conditions of interval- or ratio-level variables with normal distributions, the greater the opportunity for correlations to reach high values if the underlying associations are linear and strong.  With ordinal variables and distributions that are far from normal, correlations will be attenuated:  underlying relationships that are linear and strong will not show up as quite so strong in the observed coefficient.  Point-biserial correlations, which connect interval with binary variables, will also suffer from this sort of attenuation.  There just isn't as much opportunity for the two variables to co-vary when, for every small increment of the one measured on an interval scale, the other either remains a 0 or jumps to 1.
A: Two small points:
1) As mentioned before - the two correlation measures are measuring different things, so a strict comparison of the two is risky.
2) If you also want to make a formal inference of the difference between the two, you will probably need to go with a permutation test, since (to my knowledge) there is no formal test that will make this comparison.
