Yes, I think you may be quite right. Most of the products that I react to are quite strongly scented (the exception being Haslinger Sandalwood). I do have Proraso Sensitive Soap (white tub), and I have never had any trouble with it at all - as a matter of fact, it is my recovery soap when something else has caused me problems.Common problem ingredients include fragrance oils, essential oils and lanolin. You many want to try an unscented soap with a short ingredient list. For example, Soap Commander Integrity, WSP Au Natural, Haslinger Sensitive.
I will certainly look into unscented soaps. Thanks.