  Lorentz symmetry (LS), one of the most fundamental physical symmetries, has
been extensively studied in the context of quantum gravity and unification
theories. Many of these theories predict a LS violation, which could arise from
the discreteness of spacetime, or extra dimensions. Standard-model extension
(SME) is an effective field theory to describe Lorentz violation whose effects
can be explored using precision instruments such as atomic clocks and
gravitational-wave (GW) detectors. Considering the pure-gravity sector and
matter-gravity coupling sector in the SME, we studied the leading
Lorentz-violating modifications to the time delay of light and the relativistic
frequency shift of the clock in the space-based GW detectors. We found that the
six data streams from the GW mission can construct various combinations of
measurement signals, such as single-arm round-trip path, interference path,
triangular round-trip path, etc. These measurements are sensitive to the
different combinations of SME coefficients and provide novel linear
combinations of SME coefficients different from previous studies. Based on the
orbits of TianQin, LISA, and Taiji missions, we calculated the response of
Lorentz-violating effects on the combinations of the measurement signal data
streams. Our results allow us to estimate the sensitivities for SME
coefficients: $10^{-6}$ for the gravity sector coefficient $\bar{s}^{TT}$,
$10^{-6}$ for matter-gravity coupling coefficients
$(\bar{a}^{(e+p)}_{\text{eff}})_{T}$ and $\bar{c}^{(e+p)}_{TT}$, and $10^{-5}$
for $(\bar{a}^{n}_{\text{eff}})_{T}$ and $\bar{c}^{n}_{TT}$.
