The Natural Hair pin is a named variation usually used to describe the 1-3 of Natural Turn, Back pivot and Curved Feather.

Slow foxtrot
Man - Forward right
Side Left
Back Right
Back left foot pivoting action to right
Forward right
Forward left Left shoulder lead
Forward Right in CBMP outside partner
The last three steps curving strongly to right

Lady- Back left
Close right to left (Heel turn)
Forward left
Forward right (Pivoting Action)
Back Left
Back Right
Back Left partner outside

Waltz
Man - Forward right
Side Left
Closes Right to left
Back left foot pivoting action to right
Forward right
Forward left Left shoulder lead
Forward Right in CBMP outside partner
The last three steps curving strongly to right

Lady- Back left
Side right
Close left to right
Forward right
Back Left
Back Right
Back Left partner outside

The footwork is
Slow Foxtrot Man HT T TH THT HT T TH
Lady TH HT TH HT TH TH TH

Waltz Man HT T TH THT HT T HT
Lady TH T TH HT TH T TH

Timing
Slow Foxtrot SQQ&SQQ
Waltz 123 1&23

Janet Chapman Fellow IDTA Fellew & Examiner UKA

In American Style Foxtrot there are (at least) two hairpins. The DVIDA Silver syllabus has a Hairpin from Reverse (1-3 of an Open Left Turn followed by a back check and two side and forward steps putting the lady into a hairpin) and a Hairpin from Promenade Position (essentially a strongly curved Feather starting in PP). You can see the details in the DanceVision video (my version is with the Marantos; DanceVision probably has a newer video now) or syllabus book.

A Hairpin Feather is a Foxtrot movement. We were taught this way. After a Reverse Turn a Curving Three Step which finishes with the mans L.F. forward. Almost on the spot Back R.F. L.F. to the side R.F. forward ( foot position only ) Which is the same as any Feather Finish . The whole move is done on a Hairpin ( small). From there we were taught a Chasse into a kind of Drag Hesitation to finish diag to centre. You could if you wished simply do an Open Imputus after the Hairpin. Any one who knows how to do a Hover Cross would have no difficulty with the Hairpin