There's an interpretation given in some work on copulas.

e.g. see p 15 of [Embrechts et al](http://www.risklab.ch/ftp/papers/DependenceWithCopulas.pdf) (2001) [1], which has for the Spearman correlation of $(X,Y)^T$:

$\rho_S(X,Y)=3(\mathbb{P}\{(X-\tilde{X})(Y-Y')>0\}-\mathbb{P}\{(X-\tilde{X})(Y-Y')<0\})$

where $(X, Y)^T$, $(\tilde{X},\tilde{Y})^T$
and $(X',Y')^T$ are independent copies. (It then goes on to show your interpretation holds for that definition.)

[1] Paul Embrechts, Filip Lindskog and Alexander McNeil (2001),  
"Modelling Dependence with Copulas and Applications to Risk Management"   
http://www.risklab.ch/ftp/papers/DependenceWithCopulas.pdf  
([alternative link](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.69.792&rep=rep1&type=pdf))