This paper introduces robust twoblock (RTB) simultaneous dimension reduction, which is the first statistically robust method to perform simultaneous dimension reduction in two blocks of variables that allows to tune the number of components in each block individually. The paper proposes both a dense and a sparse version of the new method. Sparse RTB is the first robust estimator of this kind that allows to tune both the number of components and the degree of sparsity (i.e.\ the number of variables retained) for each block individually. RTB thereby allows to optimally extract and summarize the relevant portion of information in each block of data, also in the presence of outliers. As a corollary, the estimators can be recombined into a single estimate of regression coefficients for multivariate regression that is operable when the number of variables exceeds the number of cases in each block. An extensive simulation study illustrates that the new methods are resistant to different types of outliers, while maintaining estimation efficiency across a range of dimensionality settings. These findings both hold true for the dense and the sparse method. The methods' performance is further illustrated on two example data sets and a straightforward algorithm is presented and made accessible in an open source repository.

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