Is the Buddhist path one of 'selfless offering' or 'inner kindness'? Just expanding on this answer is the Buddhist path one of 'selfless offering of oneself and efforts' or of 'inner kindness' i.e. kindness to oneself. Or is it both or neither. I've come across both themes and both seem right but to me they contradict.

Of course Buddhist is about many other things including outer kindness (to all beings) but right now I'm interested in these two aspects.

Many thanks as always
The two contradict only to a "confused" (=normal) mind - in which "self" and "others" are two separate things. To the enlightened mind, what's good for one is good for the other, because they dependently-co-emerge.

Even in a regular worldly sense, if you think about it, it's impossible to 100% neglect one and only care about the other. If you attempted to completely stop taking care of yourself and only help others, you'd die - and how do you help then? If you attempted to only take care of yourself at the expense of the others you'd soon run into problems with society and the law, and with the nature at the bigger scale.

Sedaka Sutta SN 47.19:


  Looking after oneself, one looks after others.
  Looking after others, one looks after oneself.
  
  And how does one look after others by looking after oneself?
  By practicing, by developing, by doing a lot.
  And how does one look after oneself by looking after others?
  By patience, by non-harming, by loving kindness, by caring.
  (Thus) looking after oneself, one looks after others;
  and looking after others, one looks after oneself.