We present detailed JHK surface photometry with ellipse fits of 13 gal
axies selected from previous optical observations as likely candidates
for having a secondary bar or a triaxial bulge within the primary bar
. We have found 7 double-barred galaxies, 3 double-barred galaxies wit
h an additional intermediate structure with twisted isophotes, and 3 g
alaxies with a bar and central twisted isophotes. A global analysis of
the structural parameter characteristics in the I- and K-bands is pre
sented. Various numerical models of galaxies with bars within bars are
also analysed using the ellipse fitting technique and compared to the
observations. A thorough review of the possible hypotheses able to ex
plain this phenomenon is given with emphasis on the most likely ones.