What's the purpose of crossing over in the middle of a toroid winding?
This is an RF split-wound balun. Moving the windings apart is intended to reduce capacitance between input and output. As this says:


  Some articles and handbooks show a Split winding method. This method is supposed to reduce winding capacitance by moving the ends of windings further apart. The proposed theory is by reducing shunt capacitance that "leaks RF around the balun", balun performance is enhanced. 


You can find a link to another page from the same author comparing measurements to conventional winding here.