  A repulsive Hubbard model with both spin-asymmetric hopping (${t_\uparrow\neq
t_\downarrow}$) and a staggered potential (of strength $\Delta$) is studied in
one dimension. The model is a compound of the mass-imbalanced (${t_\uparrow\neq
t_\downarrow}$, ${\Delta=0}$) and ionic (${t_\uparrow = t_\downarrow}$,
${\Delta>0}$) Hubbard models, and may be realized by cold atoms in engineered
optical lattices. We use mostly mean-field theory to determine the phases and
phase transitions in the ground state for a half-filled band (one particle per
site). We find that a period-two modulation of the particle (or charge) density
and an alternating spin density coexist for arbitrary Hubbard interaction
strength, ${U\geqslant 0}$. The amplitude of the charge modulation is largest
at ${U=0}$, decreases with increasing $U$ and tends to zero for
${U\rightarrow\infty}$. The amplitude for spin alternation increases with $U$
and tends to saturation for ${U\rightarrow\infty}$. Charge order dominates
below a critical value $U_c$, whereas magnetic order dominates above. The
mean-field Hamiltonian has two gap parameters, $\Delta_\uparrow$ and
$\Delta_\downarrow$, which have to be determined self-consistently. For
${U<U_c}$ both parameters are positive, for ${U>U_c}$ they have different
signs, and for ${U=U_c}$ one gap parameter jumps from a positive to a negative
value. The weakly first-order phase transition at $U_c$ can be interpreted in
terms of an avoided criticality (or metallicity). The system is reluctant to
restore a symmetry that has been broken explicitly.
