Clearly there are a number of parts of this poem that refer to the concept of integrity. Kipling has created in this blueprint his vision of a perfect man, and thus part of this necessarily involves the discussion of integrity at various stages. We can see how this comes up in the first stanza with repeated references made to how you interact with others and how this influences you:
IF you can keep your head when all about you
Are losing theirs and blaming it on you,
If you can trust yourself when all men doubt you,
But make allowance for their doubting too;
If you can wait and not be tired by waiting,
Or being lied about, don't deal in lies,
Or being hated, don't give way to hating,
And yet don't look too good, nor talk too wise:
This stanza urges the audience to stay true to their inner integrity and not turn away from that. The listener is not to fall into the same mistakes as those that accuse them. If they are being lied about for example, they are not to "deal in lies" either. Likewise, even if everybody else doubts you, the listener has to hold true to their own integrity and self-belief, even if everybody else is questioning that.
A lot of the poem really talks about the concept of integrity and being true to ourselves. There are obviously many challenges that "men" face during life that threaten this sense of inner integrity, and throughout the poem, Kipling argues that to be a man is always to maintain that sense of integrity and not to yield on it for one moment. Note how the second stanza talks about not letting dreams or our thoughts or Triumph and Disaster change us for the worse:
If you can dream - and not make dreams your master;
If you can think - and not make thoughts your aim;
If you can meet with Triumph and Disaster
And treat those two impostors just the same;
The overwhelming message of this poem then seems to be that whatever we do, whatever is done to us, and whatever we meet in the course of life, we must stay true to who we know ourselves to be and not be influenced or dominated by any of the many possible things that could cause us to compromise on that.