= Solution
Remove the ball interiors. Each remaining piece inherits an <orientation of a topological manifold>[orientation of a manifold] and hence a boundary <orientation>. Choose the <orientation> on the second piece, reversing its original choice if necessary, so that the gluing map identifies these boundary orientations oppositely. Its two possible <orientation> signs are accommodated by this choice.
Near a glued point, the two half-space collar charts combine to give a full Euclidean chart. The opposite induced boundary orientations make the two interior orientations agree in this chart. Away from the seam, keep the orientations already chosen on the two pieces. These fit together globally, so \b[the <connected> sum is orientable]. This does not require that an arbitrarily chosen gluing preserve both initially specified orientations unchanged.
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