HI kinematic asymmetries are common in late-type galaxies irrespective of environment, although the amplitudes are strikingly low in isolated galaxies. As part of our studies of the HI morphology and kinematics in isolated late-type galaxies we have chosen several very isolated galaxies from the AMIGA sample for HI mapping. Here we present GMRT 21-cm HI line mapping of CIG 340 which was selected because its integrated HI spectrum has a very symmetric profile, Aflux = 1.03. Optical images of the galaxy hinted at a warped disk in contrast to the symmetric integrated HI spectrum profile. Our aim is to determine the extent to which the optical asymmetry is reflected in the resolved HI morphology and kinematics. GMRT observations reveal significant HI morphological asymmetries in CIG 340 despite its overall symmetric optical form and highly symmetric HI spectrum. The most notable HI features are: 1) a warp in the HI disk (with an optical counterpart), 2) the HI north/south flux ratio = 1.32 is much larger than expected from the integrated HI spectrum profile and 3) a ~ 45 (12 kpc) HI extension, containing ~ 6% of the detected HI mass on the northern side of the disk. We conclude that in isolated galaxies a highly symmetric HI spectrum can mask significant HI morphological asymmetries. The northern HI extension appears to be the result of a recent perturbation (10^8 yr), possibly by a satellite which is now disrupted or projected within the disk. This study provides an important step in our ongoing program to determine the predominant source of HI asymmetries in isolated galaxies. For CIG 340 the isolation from major companions, symmetric HI spectrum, optical morphology and interaction timescales have allowed us to narrow the possible causes the HI asymmetries and identify tests to further constrain the source of the asymmetries.