Structurally analogous proteins do exist!

By S. Krishna, Nick V. Grishin. 2004

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Comprehensive structural classification of ligand-binding motifs in proteins. (2008, background): “…irrespective of sequence similarity, sequence order, and protein fold (as currently defined) may be a useful approach to elucidating the evolutionary history of fold changes (Grishin, 2001a; Krishna & Grishin, 2004; Andreeva & Murzin, 2006; Taylor, 2007; Goldstein, 2008; Xie & Bourne, 2008).”
HorA web server to infer homology between proteins using sequence and structural similarity (2009, background): “Such ‘analogous’ similarity often comes from convergence to similar structures due to a limited number of energetically favorable ways to pack secondary structural elements (SSEs) (8, 9 ).”
Discrimination between distant homologs and structural analogs: lessons from manually constructed, reliable data sets. (2008, methodology): “( 2 ) Pearson's correlation coefficient (correl)”