Is it possible for a language to have both left-headed and right-headed compounds? And can one please explain this with examples and with the use of linguistic reasoning based on morphology?
Starting with your last request that the answer be based on morphology, this is, in fact, one of the problems of compounding, because it's not entirely a morphological phenomenon.
Compounding is gradient, ranging from lexical to phrasal to clausal compounds. Here are some examples:
(1) [[dark-]room] :: lexical (2) [[truck-]tire] :: phrasal (3) her [[don't-[go-[there-]]]face] :: clausal
Ex. (1) is a lexical compound because the dependent compound part (dark-) cannot be modified anymore: *very dark-room.
Ex. (2) is a phrasal compound the dependent compound part (truck-) can still be modified: military- truck- tire.
Languages differ concerning how much they prefer their compounds to be lexical or phrasal. German doesn't like phrasal compounds very much, while English is freer in this respect.
Ex. (3) is a clausal compound, the dependent compound part being the imperative clause don't go there. This type of compound is more common than the literature on compounding suggests. Some clausal compounds have also become names, the most famous being "forget-me-not", which is also a clausal compound in German "Vergiss-mein-nicht", and in Japanese "wasure-na-gusa".
An exocentric construction is one the category of which cannot be deduced from (the interaction between) its constituent categories. The most famous example of an exocentric construction is S = NP VP, the minimally independent syntactic category. However, most categories are endocentric, i.e. the category of the construction inherits the category type of one its immediate constituents. In [[red] cars], the entire NP inherits its category from the noun cars, rather than adjective red. The assumption is justified because red cars behaves, i.e. distributes, more like cars than like red.
Three types of exocentric compound are distinguished: bahuvrihi, dvandva, and a type of compound that includes a verb (Colin Fine's examples pickpocket and cutpurse are of this type). Here are some examples:
(4) skinhead, bluecollar :: _bahuvhrihi_ (5) bitter-sweet, maid-servant :: _dvandva_ (6) a. kick-ball, jump-rope :: verb + noun b. kick-back, have-not :: verb + particle c. let-down, shut-out :: participle + particle d. must-have, has-been :: auxiliary + verb
The examples in (4) are bahuvrihi, i.e. compounds the meaning of which cannot be reduced (or is difficult to reduce) to the semantic composition of the individual meanings of the constituent parts. Skinhead and bluecollar denote persons.
The examples in (5) are dvandva, i.e. compounds with two heads. It's difficult to decide which compound part is the dependent, and which the head.
The examples in (6) show different types of exocentric compounds, all involving verbs. However, all compounds in (6) are nouns. While nouns are present in (6a), they are the objects of the preceding verbs, and hence one cannot assume that this type of compound simply results from syntactic structure.
Headedness in compounds
Headedness in compounds is highly asymmetric. If a language has left-headed and right-headed compounds, then it is highly probably that one of these tendencies accounts for the vast majority of compounds in the language (my guess would be in excess of 90%). Most atypical compounds, such as the (probably left-headed) pickpocket and cutpurse (type 6a) are at the border to lexicalization, i.e. the entire compound is learned and memorized as one lexeme.
The preferred direction of headedness in compounds is inherited from syntax. The English NP has a clear word order within the NP, and the same order is present in compounds. That is, English minimal NPs are left-branching (i.e. right-headed), and hence compounds are also right-headed. When the underlying structure is not a minimal NP, then English compounds can take a right-branching, i.e. left-headed, structure:
(7) lady-in-waiting, bride-to-be
The compound heads in (7) are lady and bride. PPs and to-infinitival VPs must follow nouns, and this constellation is also reflected in compounds. I think that compounds of the type shown in (7) are significantly more common and more productive than the light fantastic, which to me neither seems overly productive, nor a compound. A bare adjective can rarely follow a noun, and if it does, we're probably dealing with a lexicalized construction. If the adjective is not bare, e.g. modified by an adverb, or if more than one adjective appears, then an appositive construction following the noun is possible:
(8) all the children [big and small], a man [tall and handsome]
Examples like those in (8) show again the proximity of compounding to syntax. A dependency-based account to compounding can be found here.