Allergy is due to body's immune system reacting to certain proteins or protein components. Allergy can be caused by anything with protein components which can trigger immune reaction. The susceptibility to various food allergies varies from person to person and same person may not be allergic to all food. Although usually when one is allergic to a food product or vegetable/fruit belonging to a group they might be having cross allergies to all other members of that group like for example foods containing salicylates as you mentioned. But even in such a case, the amount of the offending component like salicylate can vary in amount in different foods. So, some foods may not have sufficient amount of allergen to cause reaction in a person.
As I mentioned any protein component can trigger allergic reaction, other protein components of radish can be responsible in your case. Although your symptoms appear to be mild and can be easily due to seasonal variation, there is no harm in getting tested for possible food allergies. A food challenge test using suspected food item, in this case radish, can be done in hospital condition to test for food allergy. In food allergies reaction might be severe in second exposure. So it is better to be tested in hospital by a physician as anaphylaxis can occur sometimes following repeat exposure.
Hope this helps. Thank you.