Though it is mostly a rule-of-thumb and not formal in any way, a reaction is usually written in the sense of 'reactants' + 'reagents' equals products. When you mix two compounds that react together and a distinction between reagent and reactant cannot be made (for example, HCl + NaOH), no rule exists. In the example you gave, barium chloride can be used to test for the presence of sulfate, making the latter the reactant. More commonly, sulfate is added to a solution to test for the presence of barium, making sulfate the reagent. However, except maybe for some chemical formalism nazis, most chemists couldn't care less.