user11144: This just came up in my youth chess training.
A very general definition of "having the opposition", including even rare cases, is

A: All corners of the (axis-parallel) rectangle spanned by the two kings have the same color, the opponent has the move.

A more "operational" description of "having the opposition" is

B: Regardless how the opponent king walks out of opposition, you can walk in again.

Prove A=>B. (Danger, the board has edges, so you have to include in your proof that the mobility of your king may be limited!)
EDIT: Of course, see comment below, it is assumed that the board is empty except the two kings. Otherwise "opposition" might be meaningless due to inaccessibility of squares to the kings, and the more general theory of "corresponding squares" sets in.

user18809: Two kings are in direct opposition if they're on the same row or column separated by just one square. Two kings are in diagonal opposition if they're facing each other diagonally with one square in between. In both cases the player not having the turn has the opposition.
Indirect opposition is harder to define because different people will mean different things depending on context. The most practical definition is probably "a player has the indirect opposition if he can force his opponent to concede the opposition.
As you say in the question, the rectangle rule can give us a hint but squares defended by pawns can make a player lose this type of opposition.

user13674: Although not a mathematician, my answer is the following.
Consider the chessboard as having Cartesian coordinates, such as columns 0 to 7 and rows 0 to 7. Similar reasoning applies if 1 to 8 is chosen. The position of a king can be represented by a two item tuple, column first then row. Kings on opposite ends of a long diagonal would be W(0,0) and B(7,7) .
In the above formulation the kings will be in opposition if the differences between their corresponding tuple items are both even. Consider zero as even and both differences being zero is prohibited as they cannot occupy the same square. This meets the definition A in the query. For example with W(0,0) & B(2,4) and it's Black's move, White ( side not to move ) has the opposition.
If the Black king moves he must change either item by one or both. White simple changes the corresponding item ( coordinate ) or both. If the White king is in a corner he can still change each coordinate, or both, by one. This satisfies B in the query. The other king can always walk back into opposition.

11144: <p>This just came up in my youth chess training.</p>
<p>A very <a href="https://chess.stackexchange.com/questions/35349/description-of-rectangular-opposition-in-silmans-complete-endgame-course">general</a> definition of &quot;having the opposition&quot;, including even rare cases, is</p>
<ul>
<li>A: All corners of the (axis-parallel) rectangle spanned by the two kings have the same color, the opponent has the move.</li>
</ul>
<p>A more &quot;operational&quot; description of &quot;having the opposition&quot; is</p>
<ul>
<li>B: Regardless how the opponent king walks out of opposition, you can walk in again.</li>
</ul>
<p>Prove A=&gt;B. (Danger, the board has edges, so you have to include in your proof that the mobility of your king may be limited!)</p>
<p>EDIT: Of course, see comment below, it is assumed that the board is empty except the two kings. Otherwise &quot;opposition&quot; might be meaningless due to inaccessibility of squares to the kings, and the more general theory of &quot;corresponding squares&quot; sets in.</p>
 18809: <p>Two kings are in <strong>direct opposition</strong> if they're on the same row or column separated by just one square. Two kings are in <strong>diagonal opposition</strong> if they're facing each other diagonally with one square in between. In both cases the player not having the turn has the opposition.</p>
<p><strong>Indirect opposition</strong> is harder to define because different people will mean different things depending on context. The most practical definition is probably &quot;a player has the indirect opposition if he can force his opponent to concede the opposition.</p>
<p>As you say in the question, the rectangle rule can give us a hint but squares defended by pawns can make a player lose this type of opposition.</p>
 13674: <p>Although not a mathematician, my answer is the following.
Consider the chessboard as having Cartesian coordinates, such as columns 0 to 7 and rows 0 to 7. Similar reasoning applies if 1 to 8 is chosen. The position of a king can be represented by a two item tuple, column first then row. Kings on opposite ends of a long diagonal would be W(0,0) and B(7,7) .
In the above formulation the kings will be in opposition if the differences between their corresponding tuple items are both even. Consider zero as even and both differences being zero is prohibited as they cannot occupy the same square. This meets the definition A in the query. For example with W(0,0) &amp; B(2,4) and it's Black's move, White ( side not to move ) has the opposition.
If the Black king moves he must change either item by one or both. White simple changes the corresponding item ( coordinate ) or both. If the White king is in a corner he can still change each coordinate, or both, by one. This satisfies B in the query. The other king can always walk back into opposition.</p>
