In this short paper, we develop an original algorithm for the hierarchical clustering of multi-dimensional ordinal data, partially ordered as a component-wise lattice. The clustering process is designed not just as a way to group units, but to do this jointly inducing a partial order on the resulting groups. To this aim, the data are processed so as to generate a hierarchical sequence of lattices, on progressively larger clusters. To be consistent with the original order relation, the sequence is built as a path in the space of the congruences of the input lattice, through a greedy search algorithm. The algorithm is finally exemplified on data pertaining to life satisfaction in Italy.