Just looking at the basic properties of RVs in terms of correlation and covariance:
Suppose 4 assets; $A,B,C,D$ with $\rho_{X,Y}$ known $\forall X,Y \in \{A,B,C,D\}$.
Let, $U=A+B$, and $V=C+D$.
Then $\rho_{U,V} = \frac{Cov(U,V)}{\sigma_U \sigma_V} $,
where $Cov(U,V) = Cov(A+B, C+D)= Cov(A,C) + Cov(A,D) + Cov(B,C) + Cov(B,D)$,
where $Cov(X, Y) = \rho_{X,Y} \sigma_X \sigma_Y$.
and $\sigma_U = \sqrt{Var(U)} = \sqrt{\sigma_A^2 + \sigma_B^2 + 2 \sigma_A \sigma_B \rho_{AB}}$
and this boils down to:
$$\rho_{U,V} = \frac{ \rho_{A,C} \sigma_A \sigma_C + \rho_{A,D} \sigma_A \sigma_D + + \rho_{B,C} \sigma_B \sigma_C + \rho_{B,D} \sigma_B \sigma_D } { \sqrt{\sigma_A^2 + \sigma_B^2 + 2 \sigma_A \sigma_B \rho_{AB}} \sqrt{\sigma_C^2 + \sigma_D^2 + 2 \sigma_C \sigma_D \rho_{CD}}} $$
I think it is somewhat intuitive that if say asset A and asset C had much greater standard deviations than assets B and D then the result for U and V should converge to the same correlation as that for just A and C. Indeed the below formula is dominated by the terms with larger volatility, so in general I would state that you cannot tell the correlation of your baskets unless you know by which assets (which asset combinations) are dominant.