I'll toss in my $0.02 here since I found this helpful post by searching for "removing cyclical dependencies"
Yes. You can use DI to resolve circular dependencies. The first step to fixing any issue is finding it. Ninject complained about my circular dependency and threw a helpful exception at bootstrapping. Ninject found it for me and is forcing me to fix it. I could cheat and use property injection or method injection but that breaks my protection of class invariants (which is I think what you are complaining about).
So a vote for ctor injection via IoC because it will spot circular dependencies for you. It is then up to you to refactor and remove the architectural bug.
