I will use a sign convention used by Feynman. Look at the connection between the field definition and its matrix representation If the order of the differential operator and the potential were reversed, only the term with the cross product would change signs. That is what determines if a term is even or odd. In the matrix representation of a quaternion, the scalar forms the diagonal, so transposing the matrix leaves the diagonal unchanged, so it is symmetric. The 3-vector terms are antisymmetric because transposing the matrix flips the signs.
Gravity is the result of a metric, a symmetric tensor.
Electromagnetism involves an antisymmetric tensor. At least on the surface, it appears like these field have the correct symmetries for unifying gravity and electromagnetism as foreseen by Einstein. This generates 6 field equations. These are not the standard Maxwell equations, although every term in those equations is represented. The two scalar equations flip signs.