Thanks to [andre](https://mathematica.stackexchange.com/users/5467/andre)'s comment (where [this link](http://reference.wolfram.com/legacy/v5_2/book/section-2.8.7) is provided), I now see the effect of those delimiters (I tested it in _Mathematica_ 11 and also some earlier versions). When I add 2 newlines to the box representation of the cell:

    Cell[BoxData["\"\<a
    
    bc\>\""], "Input",
     CellChangeTimes->{{3.662918813714031*^9, 3.6629188530623317`*^9}}]
and switch back using `Shift+Ctrl+E`, then the cell look like this:

    "a
    
    bc"
But if I remove those delimiters:

    Cell[BoxData["\"a
    
    bc\""], "Input",
     CellChangeTimes->{{3.662918813714031*^9, 3.6629188530623317`*^9}}]
and switch back, then the cell looks like this:

    "a  bc"
It seems that the purpose of `\<`,`\>` is to delimit ranges within string literals in raw box representations of cells where newlines should be exactly preserved. It looks like they only have effect in that context and are ignored in normal input in newer versions of _Mathematica_. The legacy documentation suggests that in _Mathematica_ 5 they were significant in all string literals.

---
_Update:_ I found an old discussion on this topic:
[\[1\]](http://forums.wolfram.com/mathgroup/archive/2010/Dec/msg00428.html),[\[2\]](http://forums.wolfram.com/mathgroup/archive/2010/Dec/msg00462.html),[\[3\]](http://forums.wolfram.com/mathgroup/archive/2010/Dec/msg00483.html).


  [1]: https://mathematica.stackexchange.com/a/112601/280