  An understanding of spin frequency ($\nu$) evolution of neutron stars in the
low-mass X-ray binary (LMXB) phase is essential to explain the observed
$\nu$-distribution of millisecond pulsars (MSPs), and to probe the stellar and
binary physics, including the possibility of continuous gravitational wave
emission. Here, using numerical computations we conclude that $\nu$ can evolve
in two distinctly different modes, as $\nu$ may approach a lower spin
equilibrium value ($\nu_{\rm eq,per}$) for persistent accretion for a long-term
average accretion rate ($\dot{M}_{\rm av}$) greater than a critical limit
($\dot{M}_{\rm av,crit}$), and may approach a higher effective spin equilibrium
value ($\nu_{\rm eq,eff}$) for transient accretion for $\dot{M}_{\rm av} <
\dot{M}_{\rm av,crit}$. For example, when $\dot{M}_{\rm av}$ falls below
$\dot{M}_{\rm av,crit}$ for an initially persistent source, $\nu$ increases
considerably due to transient accretion, which is counterintuitive. We also
find that, contrary to what was suggested, a fast or sudden decrease of
$\dot{M}_{\rm av}$ to zero in the last part of the LMXB phase is not essential
for the genesis of spin-powered MSPs, and neutron stars could spin up in this
$\dot{M}_{\rm av}$-decreasing phase. Our findings imply that the traditional
way of $\nu$-evolution computation is inadequate in most cases, even for
initially persistent sources, and may not even correctly estimate whether $\nu$
increases or decreases.
