Find the Wavy Words!
05AB1E, 11 9 bytes (Thanks to Adnan)

Dg2›iÇü‹Ù


Try it online!

Wavy Cases:

0 - Decreasing Wavy  

1 - Increasing Wavy 

Not Wavy Cases:

[0,1] - Not wavy, initially decreasing, but then has an increase/equality that broke the pattern.  

[1,0] - Not wavy, initially increasing, but then has a decrease/equality that broke the pattern  

Input String - Not possible to be wavy in the first place due to length.

Explanation:

Dg2›iÇü‹Ù   # Full program

D           # Push 2 copies of input.
 g2›i       # If length is greater than 2. 
     Ç      # Push ASCII values for all characters in the string.
      ü     # Push pairwise array.
       ‹    # Vectorize 1 if negative difference, 0 if positive difference.
        Ù   # Uniquify using right most unique values first.
            # Else just print the string back, letting them know it's not valid input.