On the PSI website, the instructions say .... "Mechanism is a double twist, pen will extend twice per twist."
So, in answer to your question ...
"- The pen twists both directions to extend, is this right?"
YES, THAT'S RIGHT.
There are a few pens that have those double-twist mechanisms. . The Fat Cigar pens from Berea are like that.
PSI has a similar pen, called the Big Ben. . I don't know if it is double-twist or not but the instructions should tell you.
PSI is pretty good about telling you these things on their website, either in the ad for the pen or in the instructions.
Your pen looks very nice !!! . . I take it that you cast the blank yourself . . CONGRATULATIONS !! . . Well done !!