I think you have a basic misunderstanding reflected here, and an explanation may clear things up. Regular PADI courses are a recreational diving program, which can then be followed by a more advanced technical diving program. TDI is a more advanced technical diving program. It is roughly like going from high school to college. When you ask if PADI will recognize the TDI courses when you go for more advanced PADI training, it is like asking if your local high school will accept your college degree when you go in for more advanced high school classes.
I teach PADI courses, and I teach TDI courses. When students have completed all the regular PADI level courses, they will be eligible to go on to the TDI courses. TDI courses begin where the regular PADI courses end.
PADI does have a program equivalent to TDI, so a student who competes the regular PADI program can go on to the PADI tech program.
SDI is the TDI equivalent of PADI's regular program. The normal progress is complete basic SDI training and then take TDI training.
I think that what you really want to know is if the PADI and SDI programs are similar and equally recognized. You want to know if you can move back and forth between those agencies. Yes, they are. Both PADI and SDI are members of the WRSTC, so they have very similar standards. SDI is well recognized. You can cross over from one to the other.