This paper discovers molecular symmetry (MS) properties of conical intersections (CIs) and the related nonadiabatic coupling terms (NACTs) in molecules Which allow large amplitude motions such its torsion, in the frame of the relevant molecular symmetry group, focusing Oil groups with one-dimensional (1-d) irreducible representations (IREPs) If one employs corresponding MS-adapted nuclear coordinates, the NACTs call be classified according to those IREPs. The assignment IS supported by theorems which relate the IREPs of different NACTs to each other, and by properties of the NACTs related to the CIs. For example, planar contour integrals Of the NACTs evaluated along loops around the individual CIs are equal to +pi Or -pi, depending oil the IREP-adapted signs of the NACTs The + or - signs for the contour integrals may also be Used to define the "charges" and IREP's of the CIs. We derive various general molecular symmetry properties of the related NACTs and CIs These provide usefull applications, e.g, the discovery of all individual CIs allows one to generate, 1)), means of all molecular symmetry operations, the complete set of CIs at different symmetry-related locations Also, we show that the seams of CIs with different IREPs may have different topologies ill it specific plane of MS-adapted coordinates. Moreover, the IREPs impose symmetrical nodes of the NACTs, and this may Support their calculations by quantum chemical ab initio methods, even far away from the CIs. The general approach is demonstrated by application to an example Specifically, we Investigate the CIs and NACTs of cyclopenta-2,4-dienimine (C5H4NH) which has C-2v(M) molecular symmetry with 1-d IREP's. The results are confirmed by quantum chemical calculations, starting from the location of a CI based Oil the Longuet-Higgins phase change theorem, until it proof of self-consistency, i.e, the related symmetry-adapted NACTs fulfill quantization Riles which have been derived ill [Baer, M. Beyond Born -Oppenhenner: Electronic non-Adiabatic Coupling Terms and Conical Intersections, Wiley & Soils Inc Hoboken, NJ, 2006]