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We present \textit{Spitzer} full-orbit thermal phase curves of the hot Jupiter Qatar-1b, a planet with the same equilibrium temperature---and intermediate surface gravity and orbital period---as the well-studied planets HD 209458b and WASP-43b. We measure secondary eclipse of $0.21 \pm 0.02 \%$ at $3.6~μ$m and $0.30 \pm 0.02 \%$ at $4.5~μ$m, corresponding to dayside brightness temperatures of $1542^{+32}_{-31}$~K and $1557^{+35}_{-36}$~K, respectively, consistent with a vertically isothermal dayside. The respective nightside brightness temperatures are $1117^{+76}_{-71}$~K and $1167^{+69}_{-74}$~K, in line with a trend that hot Jupiters all have similar nightside temperatures. We infer a Bond albedo of $0.12_{-0.16}^{+0.14}$ and a moderate day-night heat recirculation efficiency, similar to HD 209458b. General circulation models for HD 209458b and WASP-43b predict that their bright-spots should be shifted east of the substellar point by tens of degrees, and these predictions were previously confirmed with \textit{Spitzer} full-orbit phase curve observations. The phase curves of Qatar-1b are likewise expected to exhibit eastward offsets. Instead, the observed phase curves are consistent with no offset: $11^{\circ}\pm 7^{\circ}$ at $3.6~μ$m and $-4^{\circ}\pm 7^{\circ}$ at $4.5~μ$m. The discrepancy in circulation patterns between these three otherwise similar planets points to the importance of secondary parameters like rotation rate and surface gravity, and the presence or absence of clouds, in determining atmospheric conditions on hot Jupiters.