Hydrogen bond comparison
In some cases, such as [F-H-F]- (biflouride), there is a symmetric hydrogen bond, with both H-F bonds equally strong.  

However, just because the two atoms bonding to the H are the same doesn't mean it is a symmetric hydrogen bond.  For example water dimer does not have a symmetrical hydrogen bond, the H is closer to one O atom than the other.  

There can be cases where the hydrogen being exactly in the middle is a potential energy minimum, and other cases where the middle is not a local minimum but there are two minima offset from the middle. 

See Symmetry of hydrogen bonds in solution for more information.