PCA is to CCA as ICA is to?

PCA looks for factors in data that maximize explained variance. Canonical correlation analysis (CCA), as far as I understand, is like an PCA but looks for a factors that maximize cross covariance between two data sets. So find pca like factors, that are common to two data sets.

Independent component analysis (ICA) is simillar to PCA, but it looks for factors that are statistically independent. Which result in to, in some way, more interpretable factors. E.g gene pathways, brain networks, parts of faces. Or you can say it would identify independent sources that are mixed to produced the data.

Is there a method, that is similar to ICA, as PCA is to CCA? So that would find independent components common to two datasets? Would the results actually make sense?

Likewise, in this paper on CCA+ICA (Sui et al., "A CCA+ICA based model for multi-task brain imaging data fusion and its application to schizophrenia"), the first (see footnote) step is to perform CCA, which yields a projection of each dataset into a low-dimensional space. If the input datasets are $X_1$ and $X_2$, each with $N$ rows=observations, then CCA yields $Z_1 = X_1W_1$ and $Z_2 = X_2W_2$ where the $Y$'s also have $N$ rows=observations. Note that the $Y$'s have a small number of columns, paired between $Y_1$ and $Y_2$, as opposed to the $X$'s, which may not even have the same number of columns. The authors then apply the same coordinate-changing strategy as is used in ICA, but they apply it to the concatenated matrix $[Z_1 | Z_2]$.