I did more on this topic: Here by employing my tensor algebra system for help.

1-degree: $u_a$
...to a 0-degree:
...with a 1-degree: $u^a u_a$

2-degree, symmetric on 1 and 2: $u_{ab} = u_{ba}$
...to a 0-degree:
...with a 1-degree: $u_{ab} v^a w^b$
...with a 2-degree symmetric on 1 and 2: $u_{ab} v^{ab}$
...with a 2-degree: $u_{ab} v^{ab}$
...to a 1-degree:
...with a 1-degree: $u_{ab} v^b$
...to a 2-degree: $u_{ab}$

2-degree: $u_{ab}$
...to a 0-degree:
...with a 1-degree: $u_{ab} v^a w^b$
...with a 2-degree symmetric on 1 and 2: $u_{ab} v^{ab}$
...with a 2-degree: $u_{ab} v^{ab}, u_{ab} v^{ba}$
...to a 1-degree:
...with a 1-degree: $u_{ab} v^b, u_{ab} v^a$
...to a 2-degree: $u_{ab}, u_{ba}$

3-degree, symmetric on 1 and 2: $u_{abc} = u_{(ab)c}$
...to a 0-degree: $ u_{(ab)c} v^a w^b x^c, u_{(ab)c} v^{ab} w^c, u_{(ab)c} v^a w^{bc} u_{(ab)c} w^{abc} $
...to a 1-degree: $ u_{(ab)c} v^a w^b, u_{(ab)c} v^a w^c, u_{(ab)c} v^{ab}, u_{(ab)c} v^{ac} $
...to a 2-degree: $ u_{(ab)c} v^a, u_{(ab)c} v^c $

....to a 3-degree: $ u_{abc} $