  A quantum game in the Eisert scheme is defined by the payoff matrix, plus
some quantum entanglement parameters. In the symmetric nonzero-sum 2x2 games,
the relevant features of the game are given by two parameters in the payoff
matrix, and only one extra entanglement parameter is introduced by quantizing
it in the Eisert scheme.
  The criteria adopted in this work to classify quantum games are the amount
and characteristics of Nash equilibria and Pareto-optimal positions. A new
classification based on them is developed for symmetric nonzero-sum classical
2x2 games, as well as classifications for quantum games with different
restricted subsets of the total strategy set. Finally, a classification is
presented taking the whole set of strategies into account, both unitary
strategies and nonunitary strategies studied as convex mixures of unitary
strategies.
  The classification reproduces features which have been previously found in
other works, like appearance of multiple equilibria, changes in the character
of equilibria, and entanglement regime transitions.
