Let be a finite generalized quadrangle of order (s,t), s≠1≠t. A spread is a set of st+1 mutually nonconcurrent lines of . A spread T of is called a spread of symmetry if there is a group of automorphisms of which fixes T elementwise and which acts transitively (and then regularly) on the points of at least one line (and then all lines) of T. De Bruyn (European J. Combin. 20 (1999) 759; Constructions and characterizations of near polygons, Ph.D. Thesis, Universiteit Gent, 2000, iv+203pp) has developed a method for constructing near polygons from spreads of symmetry of generalized quadrangles, and new spreads of symmetry would yield new near polygons. In this way, many new classes of near polygons were discovered by De Bruyn. If t=s², then only one class of generalized quadrangles is known that admit spreads of symmetry, namely the classical example , arising from a nonsingular elliptic quadric in PG(5,q), and in that case there is a unique class of spreads of symmetry. In De Bruyn and Thas (Illinois J. Math. 46 (2002) 797), the authors started to investigate nonexistence and existence of spreads of symmetry in various general classes of generalized quadrangles of order (s,s²), s>1, and several strong characterizations of were obtained. The most important problem that remained open in that context was to classify those elation generalized quadrangles of order (s,s²), s>1 and s odd, which have a spread of symmetry. In this paper, we completely solve that problem. No new near polygons arise. Our result also contributes to the classification of those generalized quadrangles having a line of elation points.