Hmmm, I guess I should use terms like palm side (instead of back of fingers) and non-palm side (instead of front of fingers) --- it seemed so obvious when I was looking at my left hand and typing what I could see
The going through palm to palm is a simplification -- it means you have to have adjusted the hands so that the lady can turn freely -- which is the most important bit. It also sort of helps in letting the handhold rotate as the lady turns so that you can more easily get back into the original "hook" shape handhold.
If you use a "halo" approach to turning you will probably go palm to palm but the lady's palm will be more vertical and the man's palm more horizontal.
Of course you have to find your own way of leading and you own handhold that lets you do that comfortably -- sometimes I use a chopstick style grip e.g. leading with longest finger, but keeping the first finger (nearest the thumb) free -- so that the lady's hand is then in held in a "v" sign. This gives you an extra lead -- e.g. on the left hand a free first finger can be used as a hint against the non-palm side of the lady's hand e.g. to show that I'm not leading a clockwise turn -- as that finger is obviously in the way

But I certainly wouldn't recommend that to anyone having problems.
SpinDr.