The growth of a Mn submonolayer on $\mathrm{Pt}(110)\text{\ensuremath{-}}(1\ifmmode\times\else\texttimes\fi{}2)$ was studied by surface x-ray diffraction. At room temperature, Mn fills in the empty rows of the clean substrate's missing row structure. At a coverage of 0.5 ML (monolayer), a $(1\ifmmode\times\else\texttimes\fi{}2)$ surface alloy is formed, with alternating Pt and Mn dense rows. Upon annealing (or depositing at a substrate temperature of about $570\phantom{\rule{0.3em}{0ex}}\mathrm{K}$), another surface alloy forms with a $(2\ifmmode\times\else\texttimes\fi{}1)$ symmetry. It exhibits mixed dense rows where Pt and Mn sites alternate, as in bulk ${\mathrm{Pt}}_{3}\mathrm{Mn}$. The top layer is corrugated for both the $(1\ifmmode\times\else\texttimes\fi{}2)$ and $(2\ifmmode\times\else\texttimes\fi{}1)$ surfaces, with Mn lying $0.19\ifmmode\pm\else\textpm\fi{}0.03$ and $0.16\ifmmode\pm\else\textpm\fi{}0.02\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ above the Pt site, respectively. A ${\mathrm{Pt}}_{3}\mathrm{Mn}$-like slab forms when annealing a 3-ML-thick Mn film. The observed symmetries are at variance with the NiMn and CuMn surfaces where $c(2\ifmmode\times\else\texttimes\fi{}2)$ arrangements were found. Theoretical calculations were performed for $(1\ifmmode\times\else\texttimes\fi{}2)$, $c(2\ifmmode\times\else\texttimes\fi{}2)$, and $(2\ifmmode\times\else\texttimes\fi{}1)$ PtMn two-dimensional (2D) alloys on Pt(110). Among them, the latter was found to be the ground state. Both the $(1\ifmmode\times\else\texttimes\fi{}2)$ and $(2\ifmmode\times\else\texttimes\fi{}1)$ surface alloys form antiferromagnetic (AF) Mn chains running in the $[1\overline{1}0]$ and [001] directions, respectively. The ordering within the surface layer switches to ferromagnetic (F) for a 5-ML-thick ${\mathrm{Pt}}_{3}\mathrm{Mn}(110)$ film albeit with a surface structure quite identical to the $(2\ifmmode\times\else\texttimes\fi{}1)$ 2D case. The magnetic moment per Mn atom at the surface is close to $4\phantom{\rule{0.3em}{0ex}}{\ensuremath{\mu}}_{B}$, in all cases, among the largest values ever found in similar metal-Mn surface alloys: it is directly related to the surface corrugation and to the Mn volume as already observed for other Mn-based surface alloys. The magnetic order, F or AF, is strongly influenced by the local chemical environment of the Mn sites.