  PDS 70 hosts two massive, still-accreting planets and the inclined
orientation of its protoplanetary disk presents a unique opportunity to
directly probe the vertical gas structure of a planet-hosting disk. Here, we
use high-spatial-resolution (${\approx}$0."1;10 au) observations in a set of CO
isotopologue lines and HCO$^+$ J=4-3 to map the full 2D $(r,z)$ disk structure
from the disk atmosphere, as traced by $^{12}$CO, to closer to the midplane, as
probed by less abundant isotopologues and HCO$^+$. In the PDS 70 disk,
$^{12}$CO traces a height of $z/r\approx0.3$, $^{13}$CO is found at
$z/r\approx0.1$, and C$^{18}$O originates at, or near, the midplane. The
HCO$^+$ surface arises from $z/r\approx0.2$ and is one of the few non-CO
emission surfaces constrained with high fidelity in disks to date. In the
$^{12}$CO J=3-2 line, we resolve a vertical dip and steep rise in height at the
cavity wall, making PDS 70 the first transition disk where this effect is
directly seen in line emitting heights. In the outer disk, the CO emission
heights of PDS 70 appear typical for its stellar mass and disk size and are not
substantially altered by the two inner embedded planets. By combining CO
isotopologue and HCO$^+$ lines, we derive the 2D gas temperature structure and
estimate a midplane CO snowline of ${\approx}$56-85 au. This implies that both
PDS 70b and 70c are located interior to the CO snowline and are likely
accreting gas with a high C/O ratio of ${\approx}$1.0, which provides context
for future planetary atmospheric measurements from, e.g., JWST, and for
properly modeling their formation histories.

Earth and Planetary Astrophysics;Solar and Stellar Astrophysics