  In this paper, we provide a Hilbert-style axiomatisation for the crisp
bi-G\"{o}del modal logic $\KbiG$. We prove its completeness w.r.t.\ crisp
Kripke models where formulas at each state are evaluated over the standard
bi-G\"{o}del algebra on $[0,1]$. We also consider a paraconsistent expansion of
$\KbiG$ with a De Morgan negation $\neg$ which we dub $\KGsquare$. We devise a
Hilbert-style calculus for this logic and, as a~con\-se\-quence of
a~conservative translation from $\KbiG$ to $\KGsquare$, prove its completeness
w.r.t.\ crisp Kripke models with two valuations over $[0,1]$ connected via
$\neg$.
  For these two logics, we establish that their decidability and validity are
$\mathsf{PSPACE}$-complete.
  We also study the semantical properties of $\KbiG$ and $\KGsquare$. In
particular, we show that Glivenko theorem holds only in finitely branching
frames. We also explore the classes of formulas that define the same classes of
frames both in $\mathbf{K}$ (the classical modal logic) and the crisp G\"{o}del
modal logic $\KG^c$. We show that, among others, all Sahlqvist formulas and all
formulas $\phi\rightarrow\chi$ where $\phi$ and $\chi$ are monotone, define the
same classes of frames in $\mathbf{K}$ and $\KG^c$.
