Rook behaviour in castling

Attached is a view of a precarious position  where a rook jumped over a checked square during a castle. I always thought this was impossible. That means a threatened King rook can avoid capture with a castle. Does that make sense? The prevented next move below was Re4-d6.

Yes, castling is legal if a rook is attacked.  Remember that the rule states that the king cannot be in check or move through an attacking square; the rook is free to castle if the rook itself is under attack.

Thank you. I did research and never saw a reference to rooks under attack during castling.......and I thought I knew everything about chess!

Alramech is correct.

Black can castle queenside if the c8, d8 and e8 squares aren't being attacked.

It's fine for the a8 and b8 squares to be under attack, since the king doesn't pass those squares (only the rook does).

occasionally you can take advantage of this rule by answering Rxb2 with 0-0-0+

There is no such thing as "check to a Rook".

Only Kings need to worry about check.

sometimes you can take advantage of the rook's ability to cross through an attacked square by answering Rxb2 with the DOUBLE ATTACK 0-0-0+ (checking the black Kd8 with your Rd1), while SIMULTANEOUSLY attacking the Rb2 with your Kc1.

I agree it happens

Thank you everybody.  FYI The next move after 0-0-0 was notable:- 19. Q-a5 N-b6. I missed the devastating move 20. Q x B+ followed by the taking of the knight on g4. Instead I played Qxa7 and went on to lose the match. 

Correct! Here's a simple example of that:

The King can only move one square at a time and therefore visits the intervening square en route to a castled position. The rook moves any number of squares and therefore doesn't. But mainly, the rook being attacked is irrelevant.