There are only three questions in a job interview, getting at strengths, motivation, and fit. See The Only Three Job Interview Questions, The New Leader's 100-Day Action Plan or the companion posts on strengths and motivation.
Interviewing for Fit
To be clear, this is an attempt to make an inherently complex and ambiguous subject simpler and more straightforward. It's worth it because poor cultural fit is the #1 stated reason for a new leader's failure. (Of course, stated and actual don't always match. See The top three excuses for onboarding failures.)
The fundamental questions an interviewer is getting at around fit are:
1) Will the organization be better off with you in it over time? (at any level)
2) Will you change us for the better? (at the leadership level)
Leverage the BRAVE framework (Behaviors, Relationships, Attitudes, Values, Environment)
Behave: What impact. The way people act, make decisions, control the business, etc.
Relate: How connect. The way people communicate with each other (including mode, manner and frequency), engage in intellectual debate, manage conflict, etc.
Attitude: How win. Strategy, posture and approach.
Values: What matters. Purpose and principles. This is often the critical pivot point as it gets at why people do what they do. People follow engaging leaders for awhile, but they commit themselves to a meaningful and rewarding purpose or cause over time.
Environment: Where play. External choices around markets and competition. Internal choices around layout and formality.
The suggested approach for an interviewer is to assess the interviewee's BRAVE preferences and then line them up with the organization's preferences/culture.
The suggested approach for the interviewee is to do the same thing in reverse. Follow this link for more on acing your answers to interview questions.
This is one area where no one should play any games at all. There's no upside for either the organization or the interviewee to try to be something different than what they are. They will get caught sooner or later. And the later they get caught, the more painful it will be since culture is the only truly sustainable competitive advantage.
There's more on this in my article on Analyzing Cultural Fit in Human Resource Executive.