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

e.g. see p 15 of Embrechts et al (2001) [1], which has:

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

[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)

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

e.g. see p 15 of Embrechts et al (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)