The prerequisite for meaningful comparisons of educational inequality indicators across immigration status is the comparability of socioeconomic status (SES) measures. The Programme for International Student Assessment (PISA) uses its index of economic, social, and cultural status (ESCS) to provide insights into the problems of inequality across students' socioeconomic and immigration statuses. However, missing evidence regarding the comparability of the ESCS index or its components across students with and without immigrant background challenges the accuracy of empirical inferences. Our study sheds light on the comparability of the index of household possessions (HOMEPOS) across immigration status in Norway and Sweden--two countries that continue to be two largest recipients of immigration flows among their Nordic neighbours. We tested the PISA 2018 HOMEPOS scale for the overall measurement invariance and possible differential item functioning (DIF) across three student groups with first-generation, second-generation, or no immigrant background. Several HOMEPOS items exhibited DIF within these countries. Moreover, we examined how four strategies to deal with DIF items may affect the inferences regarding educational inequalities across immigration status. The strength of the HOMEPOS--achievement association was sensitive to the choice of approach for 15-year-old immigrant students, while it remained stable and moderate for native students. Our findings encourage researchers using the HOMEPOS scale to consider the invariance testing to avoid measurement bias and provide robust evidence characterizing immigrant achievement gaps.