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If you take a commuting square of maps and use homotopy cofibers to extend outward to a big grid, you eventually run into some anticommuting squares, which are really supercommuting squares. I think this answers your last question in the affirmative. Whether this answers your other questions really depends on your standards, I suppose.

The anticommuting square is mentioned in Faisceaux Pervers and possibly also the triangulated category chapter in Weibel's Homological Algebra.

Edit: I think there is a way to write a Mayer-Vietoris sequence as chains on the totalization of a big grid like this. Then you need the signs to make it work. Here is an attempt at a diagram:

$A \to B \to B/A \to \Sigma A$

$\downarrow \qquad \downarrow \qquad \downarrow \quad \qquad \downarrow$

$C \to D \to D/C \to \Sigma C$

$\downarrow \quad \qquad \downarrow \quad \qquad \downarrow \qquad \downarrow$

$C/A \to D/B \to X \to \Sigma(C/A)$

$\downarrow \quad \qquad \downarrow \quad\qquad \downarrow \qquad \qquad \downarrow$

$\Sigma A \to \Sigma B \to \Sigma(B/A) \to \Sigma^2 A$

The bottom right square anticommutes, but the rest commute. The maps on the bottom and right edges have minus signs.

If you take a commuting square of maps and use homotopy cofibers to extend outward to a big grid, you eventually run into some anticommuting squares, which are really supercommuting squares. I think this answers your last question in the affirmative. Whether this answers your other questions really depends on your standards, I suppose.

The anticommuting square is mentioned in Faisceaux Pervers and possibly also the triangulated category chapter in Weibel's Homological Algebra.

Edit: I think there is a way to write a Mayer-Vietoris sequence as chains on the totalization of a big grid like this. Then you need the signs to make it work. Here is an attempt at a diagram:

$$\require{AMScd}\begin{CD} A @>>> B @>>> B/A @>>> \Sigma A \\ @VVV @VVV @VVV @VVV \\ C @>>> D @>>> D/C @>>> \Sigma C \\ @VVV @VVV @VVV @VVV \\ C/A @>>> D/B @>>> X @>>> \Sigma(C/A) \\ @VVV @VVV @VVV @VVV \\ \Sigma A @>>> \Sigma B @>>> \Sigma(B/A) @>>> \Sigma^2 A \\ \end{CD}$$

The bottom right square anticommutes, but the rest commute. The maps on the bottom and right edges have minus signs.

If you take a commuting square of maps and use homotopy cofibers to extend outward to a big grid, you eventually run into some anticommuting squares, which are really supercommuting squares. I think this answers your last question in the affirmative. Whether this answers your other questions really depends on your standards, I suppose.

The anticommuting square is mentioned in Faisceaux Pervers and possibly also the triangulated category chapter in Weibel's Homological Algebra.

Edit: I think there is a way to write a Mayer-Vietoris sequence as chains on the totalization of a big grid like this. Then you need the signs to make it work. Here is an attempt at a diagram:

$\begin{smallmatrix}{ccccccc} A & \to & B & \to & B/A & \to & \Sigma A \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ C & \to & D & \to & D/C & \to & \Sigma C \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ C/A & \to & D/B & \to & X & \to & \Sigma(C/A) \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ \Sigma A \to \Sigma B \to \Sigma(B/A) \to \Sigma^2 A \end{smallmatrix}$

If you take a commuting square of maps and use homotopy cofibers to extend outward to a big grid, you eventually run into some anticommuting squares, which are really supercommuting squares. I think this answers your last question in the affirmative. Whether this answers your other questions really depends on your standards, I suppose.

The anticommuting square is mentioned in Faisceaux Pervers and possibly also the triangulated category chapter in Weibel's Homological Algebra.

Edit: I think there is a way to write a Mayer-Vietoris sequence as chains on the totalization of a big grid like this. Then you need the signs to make it work. Here is an attempt at a diagram:

$A \to B \to B/A \to \Sigma A$

$\downarrow \qquad \downarrow \qquad \downarrow \quad \qquad \downarrow$

$C \to D \to D/C \to \Sigma C$

$\downarrow \quad \qquad \downarrow \quad \qquad \downarrow \qquad \downarrow$

$C/A \to D/B \to X \to \Sigma(C/A)$

$\downarrow \quad \qquad \downarrow \quad\qquad \downarrow \qquad \qquad \downarrow$

$\Sigma A \to \Sigma B \to \Sigma(B/A) \to \Sigma^2 A$

The bottom right square anticommutes, but the rest commute. The maps on the bottom and right edges have minus signs.

If you take a commuting square of maps and use homotopy cofibers to extend outward to a big grid, you eventually run into some anticommuting squares, which are really supercommuting squares. I think this answers your last question in the affirmative. Whether this answers your other questions really depends on your standards, I suppose.

The anticommuting square is mentioned in Faisceaux Pervers and possibly also the triangulated category chapter in Weibel's Homological Algebra.

If you take a commuting square of maps and use homotopy cofibers to extend outward to a big grid, you eventually run into some anticommuting squares, which are really supercommuting squares. I think this answers your last question in the affirmative. Whether this answers your other questions really depends on your standards, I suppose.

The anticommuting square is mentioned in Faisceaux Pervers and possibly also the triangulated category chapter in Weibel's Homological Algebra.

Edit: I think there is a way to write a Mayer-Vietoris sequence as chains on the totalization of a big grid like this. Then you need the signs to make it work. Here is an attempt at a diagram:

$\begin{smallmatrix}{ccccccc} A & \to & B & \to & B/A & \to & \Sigma A \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ C & \to & D & \to & D/C & \to & \Sigma C \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ C/A & \to & D/B & \to & X & \to & \Sigma(C/A) \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ \Sigma A \to \Sigma B \to \Sigma(B/A) \to \Sigma^2 A \end{smallmatrix}$