The gradient of the scalar field is a covariant vector, because of the work of the Del operator in the divisor makes it larger. The curl at first glance might appear as a draw, but it is a covariant vector capacity because of the right-angle nature of the cross product. Note that if time where to shrink exactly as much as space, nothing in the tangent quaternion would change. A quaternion equation must generate the same collection of tensors on both sides. The axial vector is the one that flips signs if the order is reversed.
Terms can continue to get more complicated.
This is called a pseudo-scalar, because it does not change under a rotation, but it will change signs under a reflection, due to the cross product. You can convince yourself of this by noting that the cross product involves the sine of an angle and the dot product involves the cosine of an angle. Neither of these will change under a rotation, and an even function times an odd function is odd. If the order of quaternion triple product is changed, this scalar will change signs for at each step in the permutation. It has been my experience that any tensor in physics can be expressed using quaternions.