We develop a new analysis approach towards identifying related radio components and their corresponding infrared host galaxy based on unsupervised machine learning methods. By exploiting Parallelized rotation and flipping INvariant Kohonen maps (pink), a self-organizing map (SOM) algorithm, we are able to associate radio and infrared sources without the a priori requirement of training labels. We present an example of this method using 894 415 images from the Faint Images of the Radio-Sky at Twenty centimeters (FIRST) and Wide-field Infrared Survey Explorer (WISE) surveys centred towards positions described by the FIRST catalogue. We produce a set of catalogues that complement FIRST and describe 802 646 objects, including their radio components and their corresponding AllWISE infrared host galaxy. Using these data products, we (i) demonstrate the ability to identify objects with rare and unique radio morphologies (e.g. 'X'-shaped galaxies, hybrid FR I/FR II morphologies), (ii) can identify the potentially resolved radio components that are associated with a single infrared host, (iii) introduce a 'curliness' statistic to search for bent and disturbed radio morphologies, and (iv) extract a set of 17 giant radio galaxies between 700 and 1100 kpc. As we require no training labels, our method can be applied to any radio-continuum survey, provided a sufficiently representative SOM can be trained.