The existence of allocations that are simultaneously fair and efficient is a central inquiry in fair division. Prominent results show that these complementary desiderata can be achieved together for indivisible items with nonnegative values. In particular, for indivisible goods and additive valuations, there always exists an allocation that is both envy-free up to one item (EF1) and fractionally Pareto efficient (fPO). An analogous EF1 and fPO guarantee is now known for indivisible chores. For indivisible mixed manna, the compatibility of fairness and efficiency is more delicate. Indeed, recent work shows that EF1 and fPO are incompatible in the additive mixed-manna setting. Through principled relaxations of EF1, we show that compatibility with fPO can be restored even for indivisible mixed manna under additive valuations. For indivisible mixed manna and additive valuations, we establish the existence of allocations that are fPO and introspectively envy-free up to one item (IEF1). In an IEF1 allocation, each agent can eliminate its envy toward every other agent by either adding an item to or removing an item from its own bundle. The notion of IEF1 coincides with EF1 for indivisible chores, and hence our result generalizes the existence guarantee for chores. We also prove the existence of allocations that are fPO and satisfy a chiral version of IEF1, which we call extrospectively envy-free up to one item (XEF1). This result generalizes the existence guarantee of EF1 and fPO allocations for goods, since XEF1 is equivalent to EF1 for goods. Finally, we utilize our IEF1 and fPO result for indivisible items to provide a distinct proof of the known EF and fPO guarantee for divisible mixed manna. Our results underscore the power of the Knaster--Kuratowski--Mazurkiewicz Theorem for proving the existence of fair and efficient allocations of indivisible and divisible items.

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