A RETURN AND LAND IS MANDATED WHEN THE CREW OF A B757-200 HAS A PROB WITH THEIR RUDDER TRIM IN CRUISE AND THEIR SPOILER SYS ON DSCNT AND LNDG AT LAX; CA.

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MY FO WAS HAND FLYING IN CLB AND NEEDED 2 UNITS OF L RUDDER TRIM. AT LEVELOFF; HE TOLD ME THAT HE COULD NOT TRIM TO THE R. I WATCHED HIM TRIM AND THE TRIM WAS STICKING ANYWHERE FROM 2-5 SECONDS MAX. THIS HAPPENED MORE THAN 3 TIMES. I TOLD FO TO FLY AND DO RADIOS. I CALLED MAINT COORDINATOR; WHO TOLD US THE ACFT HAD HISTORY ABOUT THE RUDDER TRIM. OUR ACFT ALSO HAD EGT PROBS WHICH WERE CHRONIC. WE TALKED ABOUT THE PROB AND DECIDED THAT DUE TO THE DIFFICULTY OF COM; NIGHT IN MEXICO; LIMITED DIVERSION STATIONS; TSTMS IN DEST AND IF AN ENG FAILURE WE WOULD HAVE GREATER PROBS; SO WE WENT TO LAX. I ADVISED MAINT COORDINATOR. THE ACFT FLT MANUAL HAD NO REF TO RUDDER TRIM ISSUES. THERE WERE NO EICAS MESSAGES. I TOLD MAINT COORDINATOR WE WOULD BE OVERWT FOR THE LNDG. FO GOT CLRNC BACK TO LAX OVER SANFE INTXN. DISCUSSED SAN AS ALTERNATE; BUT DUE TO SHORT RWY AND NPA; LAX AS BEST DECISION. SOCAL GAVE US RWY 25L ILS. ON DSCNT; I TOOK ACFT TO LNDG. I DECLARED THE EMER DUE TO OVERWT LNDG. WITH AUTOPLT ON AND SPD BRAKES IN USE; THE YOKE DEFLECTED 3 DEG UNITS OF AILERON MARKINGS TO THE R. I RETRACTED THE SPD BRAKES AND THE YOKE RETURNED TO NORM. I DIDN'T WANT TO USE THE SPD BRAKES AGAIN FOR REST OF FLT. NO EICAS FOR THAT AS WELL. WITH LAX RWY 25L IN SIGHT; I DISCONNECTED THE AUTOPLT AND HAND FLEW TO THE RWY. TOUCHDOWN WAS SMOOTH; BANK ANGLES WERE LESS THAN 30 DEGS. ON ROLLOUT; THE SPD BRAKES EXTENDED 1/2; THEN THE NOSE ABRUPTLY PITCHED UP. I IMMEDIATELY PUSHED THE NOSE OVER AND GNDED THE NOSEWHEEL SMOOTHLY. NO OTHER PROBS OCCURRED. ATC WANTED NO INFO FROM US. AT GATE; I CALLED FLT OPS AND TOLD THEM WHAT HAPPENED. WROTE UP ACFT THEN WENT TO OTHER ACFT AND CONTINUED W. SUPPLEMENTAL INFO FROM ACN 619582: THE RUDDER TRIM DID NOT RESPOND PROPERLY AS PLT ATTEMPTED TO READJUST IT. THE RUDDER TRIM ONLY RESPONDED AFTER HOLDING THE TRIM KNOB TO THE R SEVERAL SECONDS.