What is an Awry Grid?

If a 4×4 grid (where each cell contains an arrow in an orthogonal direction) conforms to a special rule, I call it an Awry Grid. If it does not conform to this rule, I call it a Aright Grid.

Here are some examples (click on an image to view larger version):

For a full answer, you must figure out two things:

1. A rule to determine if a given grid is awry or aright
2. A way to make each awry* grid into an aright grid, only by changing the value of a single cell within the grid, which is not the first cell marked by red (0,0). Once you figure out 1), there might seem to be multiple ways to do this, at first. But for each grid, there is only one correct operation in order to achieve this. Finding this may require digging a little bit deeper into the structure of the grids.

(*) There are special cases of awry grids, I call them Lost Grids. You cannot (properly) turn a lost grid into an aright grid. There are no cases of lost grids in this puzzle, but this note might serve as a little hint.

---

Hint #1:

Hint #2 (fairly big):

---

Notes:

• You can easily make awry and aright grids using other specifications than 4×4 + four different arrows. This is simply the configuration I chose for this puzzle
• The enigmatic-puzzle tag is there because I deliberately left out a specific tag in hopes of making this not too easy.
• The graphical elements are from here and here.