Allergen-free
In practice the phrase usually means the brand has excluded a particular list: most often the fragrance allergens that regulations require to be declared by name, and sometimes other common sensitisers such as lanolin, certain preservatives or specific plant extracts. Which list was used is rarely stated on the pack, so the same two words can mean quite different things on two different boxes.
"Hypoallergenic" behaves the same way. It suggests a lower likelihood of reaction rather than proof of one, and there is no single standardised test behind it. Neither claim can tell you whether your particular trigger is in the formula.
If you have a diagnosed contact allergy, the ingredient list is far more reliable than the front of the pack — look for your own allergen under its INCI name. Patch test anything new on a small area for several days, introduce one product at a time, and take reactions that are severe, spreading or persistent to a dermatologist or allergist.