= Solution
A <Landau free energy> is a coarse-grained thermodynamic potential expressed as a function of a proposed macroscopic <order parameter>. Its coefficients include microscopic and thermal effects already averaged over. At fixed external thermodynamic conditions, equilibrium selects its <global minimum>; other <local minima> can represent <metastable> states. It is not the internal energy of a single microscopic magnetic configuration. Here the real scalar $m$ represents a selected component of <magnetization>: zero describes the disordered state, while nonzero values distinguish two oppositely ordered states.
Every power is even, so the internal <discrete symmetry> is
$$
\boxed{m\longmapsto-m,\qquad \mathbb Z_2.}
$$
For a magnetic <order parameter>, this represents reversal of the magnetization, as under <time reversal> in zero applied <magnetic field>. A nonzero field would generally add an odd term and break this symmetry explicitly. This scalar model has no continuous internal rotation symmetry; physical rotational symmetry of a vector magnet is not implied by treating one real component. The condition $c>0$ makes the polynomial bounded below at large $|m|$.
For the requested ordered phase on cooling, take the usual implicit assumption $a>0$, and write $r=a(T-T_c)$. If $a<0$ the temperature directions reverse; if $a=0$ this expansion has no temperature-driven transition. The source explicitly states only $c>0$.
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