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The method combines bilinear decompositions of the two-way matrices with clustering over observations. Different clusterings are defined for each part of the bilinear decomposition, which decomposes the matrix-valued observations into overall means, row margins, column margins and row\u2013column interactions. Therefore up to four different classifications are defined jointly, one for each type of effect. The computational burden is greatly reduced by the orthogonality of the bilinear model, such that the joint clustering problem reduces to separate problems which can be handled independently. Three of these sub-problems are specific cases of <jats:italic>k<\/jats:italic>-means clustering; a special algorithm is formulated for the row\u2013column interactions, which are displayed in clusterwise biplots. The method is illustrated via an empirical example and interpreting the interaction biplots are discussed. 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