We test scale invariance, a prominent simplifying assumption that is often used to characterize bargaining solutions for theoretical models, in the experimental laboratory. Our study relies on a context-rich bargaining environment and varies the parameters of the bargaining problem along with the information that bargaining parties have about each other. It aims at understanding whether bargaining is guided by utilities as assumed by the classic version of cooperative bargaining theory or rather by comparisons in observables (e.g., money) as often assumed by behavioral models of decision-making. The experimental results show that scale invariance is supported when the relevant information is only privately known and when rescaling only affects the extreme (minimum and maximum) anchoring points of the utility scale. Scale invariance is not supported when rescaling affects the units on the utility scale. Overall, our experimental data deliver scarce empirical support to theoretical bargaining solutions based on unobservables like abstract utility units.