Find the Swirling Words!
MATL, 33 31 30 bytes

lydhg)dt|dZSXz&amp;=wZSdh?4M1)3M1)


Input is in uppercase letters (or in lowercase letters, but not mixed).

Output is:


If the word is non-swirling: no output is produced
If it is swirling: two numbers are produced in different lines:


First number 1/ -1  indicates centrifugal / centripetal. 
Second number 1/ `-1'  indicates clockwise / counterclockwise.



Try it online! Or verify all test cases (code modified to take all inputs and to produce the two output numbers on the same line)

Explanation

Let's take input 'OPERAA' as an example.

The first part of the code removes double letters:

l     % Push 1
      %   STACK: 1
y     % Take input implicitly from below, and duplicate
      %   STACK: 'OPERAA', 1, 'OPERAA'
d     % Convert to code points and compute differences
      %   STACK: 'OPERAA', 1, [1 -11  13 -17 0]
h     % Concatenate horizontally
      %   STACK: 'OPERAA', [1 1 -11  13 -17 0]
g     % Convert to logical
      %   STACK: 'OPERAA', [true true true true true false]
)     % Index
      %   STACK: 'OPERA'


We now check if the distances between letters are non-decreasing (necessary condition for the word to be swirling):

d     % Convert to code points and compute differences
      %   STACK: [1 -11  13 -17]
t|    % Duplicate and take absolute value
      %   STACK: [1 -11  13 -17], [1 11  13 17]
d     % Differences
      %   STACK: [1 -11  13 -17], [10 2 4]
ZS    % Signum
      %   STACK: [1 -11  13 -17], [1 1 1]
Xz    % Remove zeros (gives a vertical vector). Needed for words like 'IRIS',
      % where some consecutive distances are equal
      %   STACK: [1 -11  13 -17], [1; 1; 1]
&amp;=    % All pairwise equality comparisons. Gives a matrix. If all the signs 
      % were equal the matrix will contain all ones
      %   STACK: [1 -11  13 -17], [1 1 1; 1 1 1; 1 1 1]


We then check if the letters go back and forth  (this is the other condition for the word to be swirling):

w     % Swap
      %   STACK: [1 1 1; 1 1 1; 1 1 1], [1 -11  13 -17]
ZS    % Signum
      %   STACK: [1 1 1; 1 1 1; 1 1 1], [1 -1 1 -1]
d     % Differences
      %   STACK: [1 1 1; 1 1 1; 1 1 1], [-2 2 -2]


Lastly, we check if the two conditions hold, and in that case produce the output:

h     % Concatenate horizontally
      %   STACK: [1 1 1 1 1 1 1 1 1 -2 2 -2]
?     % If all elements are nonzero
  4M  %   Push first signum array without zeros, from the automatic clipboard
      %     STACK: [1; 1; 1]
  1)  %   Get first element (tells if first difference was positive or negative)
      %     STACK: 1
  3M  %   Push second signum array, from the automatic clipboard
      %     STACK: 1, [1 -1 1 -1]
  1)  %   Get first element (tells if first movement was right or left)
      %     STACK: 1, 1
      %   Implicitly end if
      % Implicitly display