Based on your stated requirements, you may find the software you seek, if you ensure that one of the features is mirroring.
There's two ways that a program manages mirroring. One is dynamic. In this case, you specify a centerline or centerlines and anything you create on one side is duplicated on the other. I'm currently learning a program for 3D modeling in which this is true. It's called Metasequoia, but I would not recommend it for your purposes. I've used it only as an example with which I am aware.
The "static" version of mirroring requires a manual activation of the feature. Again, the center line is created or a grid of the program is used as a reference. Once the drawing is completed, one uses the mirror feature to create a flipped version.
In Inkscape, a free multi-platform vector editor, for example, one creates the primary image, duplicates it, then mirrors that and rejoins it to the original. A few more steps than the dynamic version.
For bitmap editors, GIMP is also a free multi-purpose program which supports a mirror feature. There's a very short video showing an example of the steps described above for Inkscape, but specific to GIMP.
There are plenty of tutorial videos on the internet for both programs and for many others, should you discover another to your liking.