TITLE: In ZF, how would the structure of the cardinal numbers change by adopting this definition of cardinality?
QUESTION [10 upvotes]: In ZFC, a good way of ordering sets by cardinality is by leveraging the notion of an injection. We define: $$X \lesssim Y \leftrightarrow \mbox{ there exists an injection } X \rightarrow Y.$$
Alternatively, we can leverage the notion of a partial surjection, by which I mean a partial function that hits every point of its codomain at least once.
$$Y \succsim X \leftrightarrow \mbox{ there exists a partial surjection } Y \rightarrow X.$$
Now in ZFC, these approaches are equivalent, in the sense that $$X \lesssim Y \iff Y \succsim X.$$
But if we just consider ZF (that is, we drop choice) then the equivalence breaks down. However, we've still got the forward implication, because the converse of an injection is always a partial surjection.
A specific question. Define that two sets are weakly equipotent iff $X \succsim Y$ and $Y \succsim X$. Question: does weak equipotence imply genuine equipotence? I'm guessing "no."
A general question: In broad terms, how would the structure of the cardinal numbers in ZF be altered by adopting $\succsim$ rather than $\lesssim$ as the fundamental order relation, and taking equality of cardinal numbers to correspond to weak equipotence of sets? I'm guessing this would hide a lot of the (usually very complicated) structure of the cardinal numbers in ZF.
Remark. The order symbols can be written \lesssim and \succsim.

REPLY [4 votes]: The order which you define is commonly denoted by $\leq^*$, and I will use that notation in my answer as well.
As Andres wrote in his excellent answer, without the axiom of choice not much is provable about $\leq^*$. We know that if $\leq^*$ is antisymmetric (on cardinals) then there are no Dedekind-finite sets. We also know that if it is a total ordering then the axiom of choice holds.
We don't know a whole lot more other than that. Certain choice principles such as the partition principle (which essentially states $\leq=\leq^*$) imply the antisymmetry of $\leq^*$. 
Specific counterexamples can be found when Dedekind-finite sets exist, for example, one can have a Dedekind-finite set $A$ such that $A<A+A\leq^*A$, so trivially $A\leq^*A+A$ but because the set is Dedekind-finite we have $A\nsim A+A$.
Beyond that, one can show the following statement:

Theorem. Suppose that $V$ is a model of $\sf ZFC$. For every set $A$ there exists a symmetric extension of $V$ such that $\mathcal P(A)$ can be embedded into the cardinals ordered by $\leq^*$ and $\leq^*$. Moreover, one can preserve $\sf DC_\kappa$ for an arbitrary $\kappa$ while doing so.

One can extend this and show the even weirder statement is true:

Theorem. Suppose that $V$ is a model of $\sf ZFC$, then there is a class-symmetric extension such that $(\mathcal P(\sf Ord),\subseteq)$ can be embedded into $\leq$ and $\leq^*$, while preserving $\sf DC_\kappa$. If $V$ satisfies global choice then $V$ itself embeds into the cardinals as well.

One interesting consequence from these theorems is that $\sf DC_\kappa$ is completely orthogonal to any chaotic behavior in $\leq^*$ and $\leq$. Another interesting conclusion is that one can have a decreasing chain of cardinals of any ordinal length (although not of length $\sf Ord$) alongside $\sf DC_\kappa$.