Symmetry can often simplify the number of numbers actually needed to describe a tensor. There are a variety of important spin-offs of a standard vector. Dual vectors, when multiplied by its corresponding vector, generate a real number, by systematically multiplying each component from the dual vector and the vector together and summing the total. If the space a vector lives in is shrunk, a contravariant vector shrinks, but a covariant vector gets larger. A tangent vector is, well, tangent to a vector function.
Physics equations involve tensors of the same rank.
There are scalar equations, polar vector equations, axial vector equations, and equations for higher rank tensors. Since the same rank tensors are on both sides, the identity is preserved under a rotational transformation. One could decide to arbitrarily combine tensor equations of different rank, and they would still be valid under the transformation. There are ways to switch ranks. If there are two vectors and one wants a result that is a scalar, that requires the intervention of a metric to broker the transaction.