I’ve learned that similar sequences have similar structures.

That is not true sometimes. See this paper:

http://www.ncbi.nlm.nih.gov/pubmed/18004789

Another paper mentions a number of different tools:

http://www.ncbi.nlm.nih.gov/pubmed/27307633

In particular, the authors look at different levels of protein structure flexibility.

There are a lot of structure comparison tools:

http://molbiol-tools.ca/Protein_tertiary_structure.htm

https://toolkit.tuebingen.mpg.de/sections/tertstruct

https://www.expasy.org/proteomics/protein_structure

http://www.geneinfinity.org/sp/sp_structcompare.html

http://www.rcsb.org/pdb/workbench/workbench.do

http://bip.weizmann.ac.il/toolbox/structure/tertiary.htm

And there are much more. I think you need to look at them, they may be helpful.

See this post if you need some special structures:

How Can I Extract Only Specific Structures From Pdb