When flying an orthodrome, with few exceptions, the heading will be constantly changing. In theory the change is continuous, but in practice most aircraft are quantatized to the nearest degree.
An exception are missiles, where because of a higher speed, having a more precise heading is more critical. Therefore most longer range missiles will internally use a higher granularity in the heading (like 0.01 degree or less).
To be clear, a loxodrome is a course which crosses all meridians of longitude at the same angle, and has a constant bearing measured to true or magnetic north. Loxodrome are also called rhumb lines. All loxodromes spiral from one pole to the other pole, except longitudinal loxodromes.
An orthodrome is also called a great circle route, and is characterized with heading changes (for most headings) to allow the vessel/aircraft to fly the shortest path along the surface of the earth to get to another point on the earth. Assuming that the earth is a sphere, an orthodrome is defined by a plane which goes through the center of the sphere, and the curved lines formed by the outer portion of the sphere intersecting the plane form what is known as a great circle route. The equator and meridians of longitude and their inverse lines on the other side of the sphere, form orthodromes. In those examples, travel on the equator is a constant heading of 090 or 270. On longitude lines, the heading is either N or S, until polar passage. An equatorial orthodrome cannot be a loxodrome. A longitudinal orthodrome is a loxodrome, although a rather uninteresting one.
It all seems rather straight forward now, right? There are wikis on both rhumb lines / loxodromes and also great circle routes or orthodromes. The graphics in them may help understand things.
Addendum to readdress question from OP:
If an aircraft flies straight ahead, i.e. there is no wind, the rudder
is not actuated and the aircraft is parallel w.r.t. the surface of the
earth underneath it, will it follow an orthodrome/great circle?
The answer is always yes.
What the OP appears to be asking is whether the airplane, flying with only an inertial reference (NOT an inertial guidance system) and a fixed distance above the surface of the earth, can be accomplished without a change in direction, except for the circling of the earth.
So to explain this, let's call the earth a sphere for this discussion. If a great circle route is extended, it will scribe a line which wraps around the sphere, and divides the sphere into two identical half-spheres. Those half-spheres can be made with a single directional cut of the sphere. If you will, the sphere is split in half by a plane (geometric type, not aeronautical type) and the circle formed by that plane is the great circle route wrapping around the sphere.
The heading of an airplane flying a great circle route will with few exceptions, be constantly changing. The exceptions are when the plane is the equator or is a meridian of longitude and the corresponding reciprocal meridian.
So once again, when the path of the aircraft is constrained to the surface or some fixed distance above the surface, the inertial direction relative to the surface of the sphere, will remain a constant direction. Since only great circles which pass through the poles or travel the equator have one axis fixed relative to lat/long of the earth, they will have constant headings. The ones which are polar will have heading flips at the poles. All other orthodromes will have continuously changing headings.
There is just one more aspect of navigation that I would be remiss to not mention, and that is Transport Wander, which may be observed on a heading indicator on an aircraft, and is the function of the sin(track angle) * delta longitude/flight hours * tan(latitude)/60. The polarity changes with east vs west and northern vs southern hemispheres.
To the OP, I am sorry that I misinterpreted your question, and for the resultant confusion.