Related to this is the sense each friend gives the other of being a special individual, on whatever grounds this recognition is based.

on whatever grounds this recognition is based 等于：

whatever grounds this recognition is based on

be based on：依据；基于

whatever grounds：无论什么样的理由

whatever grounds this recognition is based on：这种认识无论基于什么样的理由

这是whatever 引导的“让步从句”，哪来的倒装？

仔细体会这些whatever 让步从句，培养语感：

We shall remain friends whatever may happen.

无论发生什么事，我们将永远是朋友。

Whatever nonsense the papers print, some people always believe it.

不管报纸胡说什么,总有一些人信以为真。

Whatever troubles arise, we'll have peace of mind amidst seeming chaos.  

无论出现什么样的麻烦，我们都会在貌似混乱的情形中保持平和的心态。