How to assign numbers to response options on a bipolar scale? Does it matter which one of the three scales we use: A, B, or C in the attached questionnaire? Each one will have a different interpretation of the data. Please advise how should we analyze the bi-polar items of this questionnaire. Thank you.
Yes, it matters, and you should probably go with scale B if you have to use numbers. See my answer here on Cross Validated in response to, "Is 0 a valid value in a Likert scale?" The zero in the middle of A is confusing (this isn't to say I don't understand it, but do you really trust that every respondent you ever come across will?), but even if you were to replace it with 4 and add 1 to the options currently labeled 4–6, you might get different answers than if you go with (-3)–(+3). Either way, there's some connotation of "less" applied to the options on the left, which may bias responses. Option C even confuses me, so I'd say you should throw that out completely...

IMO, the safest bet would be to label your rating options with words, not numbers. To make responding easy, let people check boxes or provide abbreviated codes for their preferences such as "Strongly prefer left option" = SL or "Right option is a mildly better fit" = Rm. Here's how I presented these suggestions in my other answer, and the supporting reference (Schwarz, Knäuper, Hippler, Noelle-Neumann, &amp; Clark, 1991):


  
  Allow respondents to check cells in a table corresponding to their answer preference: each row can be a different item, and each column can be labeled with your rating anchor, or vice versa – no numbers involved. Here's how that might look (if one were to answer wisely):
  
  $\begin{array}{|c|c|c|c|c|c|c|}\hline&amp;\tiny\text{Strongly Disagree}&amp;\tiny\text{Disagree}&amp;\tiny\text{Mildly Disagree}&amp;\tiny\text{Mildly Agree}&amp;\tiny\text{Agree}&amp;\tiny\text{Strongly Agree}\\\hline\tiny\text{Tumblers: better than pumpers!}^*&amp;&amp;&amp;&amp;&amp;&amp;\checkmark\\\hline\tiny\text{I look fat in this dress.}&amp;\checkmark\\\hline\end{array}$*
  
  Wikipedia gives another style using marked options (by Nicholas Smith):
  
  
  Letter codes can also be substituted for numeric options if blanks are to be filled for a list of very many items; e.g., {SD,D,MD,MA,A,SA}. Just don't forget to include the legend!
  
  
  
  
  Reference
  Schwarz, N., Knäuper, B., Hippler, H. J., Noelle-Neumann, E., &amp; Clark, L. (1991). Rating scales numeric values may change the meaning of scale labels. Public Opinion Quarterly, 55(4), 570–582.