Because every two-dimensional manifold is a connected sum of sum number of tori and some number of real projective planes, and there's a fancy relation where a torus and a RPP is actually just three RPPs, you can make a cool chart. Something like this (with the row and columns representing direct sums):

| Torus | Double torus | Triple torus | Quadruple torus | Quintuple torus |
---|---|---|---|---|---|

Real projective plane | Dyck's surface | Quintuple real projective plane | Septuple real projective plane | Nonuple real projective plane | Undecuple real projective plane |

Klein bottle | Quadruple real projective plane | Sextuple real projective plane | Octuple real projective plane | Decuple real projective plane | Duodecuple real projective plane |

Dyck's surface | Quintuple real projective plane | Septuple real projective plane | Nonuple real projective plane | Undecuple real projective plane | Tredecuple real projective plane |

Quadruple real projective plane | Sextuple real projective plane | Octuple real projective plane | Decuple real projective plane | Duodecuple real projective plane | Quattuordecuple real projective plane |

Quintuple real projective plane | Septuple real projective plane | Nonuple real projective plane | Undecuple real projective plane | Tredecuple real projective plane | Quindecuple real projective plane |

This was way more boring than I expected it to be, because after making this table I just realised that adding one cross-cap kills orientability so it's just going to be a whole bunch of crosscaps. Shrug emoji.