Identification of some real interpolation spaces

We interrelate the real interpolation spaces associated with the couples $(X,Y), (X+Y,Y), (X,X\cap Y)$, and $(X+Y,X\cap Y)$, proving among others the identities

$(X+Y,X)_{\theta,p} \cap (X+Y, Y)_{\theta,p}$	$= (X+Y, X \cap Y)_{\theta,p},$
$(X+Y,X)_{\theta,p} \cap (X+Y, Y)_{1-\theta,p}$	$= (X,Y)_{\theta,p},$
$(X,X\cap Y)_{\theta,p} + (Y,X\cap Y)_{\theta,p}$	$= (X+Y, X\cap Y)_{\theta,p},$
$(X,X\cap Y)_{\theta,p} + (Y,X\cap Y)_{1-\theta,p}$	$= (X,Y)_{\theta,p}$

for all $p \in [1,\infty], \theta \in [0,1]$.